I suspect it would look something like the black hole Gargantua in Interstellar, but instead of a dark center, it would be glowing, like a star? It would still produce gravitational lensing and bend light around it, like a black hole, yes? And if there is gas and debris in the vicinity (which, likely it would be, assuming the neutron star was created out of a supernova), it would spin around the star in an accretion disk? Any other pointers?
We believe neutron stars cool down very fast, compared to other stellar remnants, so that they are only "hot" for a comparatively short time. Therefore despite their novel phase/ state, their magnetic/ superfluidity/ possible reheating phenomenae, and possible infalling material, most neutron stars quickly become cold, dead, tiny, unobservable cinders in just a few million years. In between they would presumably be ordinary black body sources, and radiate in the visible spectrum depending on their temperature as they cool, but you'd have to be EXTREMELY close to see them as more than dots, or at all..
We can't be certain because they cease to be currently observable via light, X rays or other electromagnetic radiation within a short time, of perhaps a few hundreds of thousands or.millions of years.
There could be accretion effects, and lensing effects, but in both cases they are very limited in scale and visibility by the neutron star's tiny size - unlike black holes which can apparently be any size, neutron stars can only be tiny, tiny objects (typically 10 miles across,1.4 times the mass of our sun, maximum mass believed to be a little larger, between 2.3 and 3.0 solar masses). If they gained much more mass via accretion, collision, or other means, they'd just collapse into a black hole.
To elaborate on the answer by L. Dutch, the expected cooling is - direct Urca process cools new neutron stars down to about 1 billion K within minutes by fast neutrino loss. After that, modified URCA processes, bremsstrahlung from nucleons, particle annihilation, Cooper pairs (superconductivity/superfluidity), and so on, for a few tens or 100 thousand years. By then, the surface has reduced to about 1 million K, and photon cooling dominates, hence X ray emission dominates, shifting to a spectrum centred on less energetic wavelengths as it cools.
See this answer on astronomy SE for more.
Now, even our fairly modest sun (yellow dwarf star, 1.4 million km diameter) is a bit too hot to see close up without protection, so a tiny neutron star (10-20 km diameter) would have to be seen even closer, and cool even more. Even white and red dwarf stars can take trillions of years to cool after fusion ceases. Neutron stars are much smaller but much denser with phenomenas like superfluidity and magnetic losses to consider as well.
So yeah, you'd still either need special remote observation equipment, magical protection or you fry, for a long, long time.
However neutron stars are expected to continue to cool very fast indeed, due to their tiny size, despite their huge mass and density. This and this Q&A on physics SE suggests that within a few million, or tens of million, years at most, the temperature has already dropped to just a few thousand degrees (the temperature of the surface of the sun), and therefore looking at a universe billions of years old, "most neutron stars are cold dead cinders". A key factor is that the matter in such a star is very unsuited for retaining/containing heat, and it's tiny so there isn't much scope for heat to be retained due to being giant sized. Uncertainties exist because there is some scope for reheating effects - we simply don't know. Potentially they could be detected by their gravitational lensing effect more than by their light or EM radiation. These linked questions are worth a read.
However as they are very small, there is a lot of room for speculation, as once they cease being detectable sources at extreme temperatures, they become completely unobservable via electromagnetic waves from earth.
As we have only every observed hot young neutron stars by EM radiation (light/xray etc), the rest is a mix of best guesses, of materials in phases and states we have never directly studied in the lab.
So the best guess is very boring indeed.....
- In their initial extremely high temperature state, they would radiate via various kinds of extreme high energy. You couldn't get close enough to observe without being fried, as they are so very tiny, and if you could they wouldn't be any "colour" or glow, they'd be ferocious emitters of neutrinos, other particles, gamma rays, x rays (and some radiation across all other lower frequencies, but that pales into insignificance).
- Once photon cooling takes over at around 100k years, they are still initially ferocious "soft" X ray emitters (the soft part is the term for the X ray spectrum, its still lethal!). They remain photonic cooling emitters for the rest of their lives unless something changes (merger, impact, accretion etc)
- Photonic cooling we may assume causes them to radiate as a black body, or close to it. A black body spectrum has a peak at some frequency, and tapers off at both sides around it. As it cools, that peak moves to lower frequencies (X ray -> ultraviolet -> visible light -> infrared -> radio -> microwave) and the intensity, or power output, also falls - it becomes "dimmer".
So your neutron star won't have any special glowy effect,unless it happens to gravitationally lens something else. It'll be a tiny, tiny sphere, that looks like some colour or other for the brief time its visible, and needs to be within a tiny distance for its 10 mile size to appear as anything other than a coloured dot. After that its a cold dark cinder.
If it gains an accretion disk or passes through gassy medium, that's capable of infalling not just orbiting at distance, then it has intense gravitational energy. Black holes are extremely efficient converters of mass to energy from infalling material, something like 30% for quasars from friction effects, and neutron stars could potentially be similar. You'd be best to look up black hole accretion disk appearances for an idea. But be aware black holes are not size limited. Neutron stars are. If they gain much more mass, they'd easily cease to be a neutron star and collapse into a black hole. So the size of accretion disk is very limited, you can't get a giant NS like you can a giant BH, whether measured by size or mass, and it can't accrete forever as it will eventually collapse.
It will produce lensing effects but again, it's a tiny, tiny object, unlike many BH's, so the lensing effect won't be nearly as noticeable. However the effect will persist indefinitely, even when the NS is cold, as its not influenced by the neutron star's temperature.
You can't get closer than an orbit
The gravity of a Neutron Star is INTENSE. Just on the verge of a black hole. If a human were to stand on the surface of one, his body would exert of downward force of about 460 trillion Newtons. Not only would this kill you, but you would cease to even exist as normal matter before you get close enough to make a visual inspection of the star's surface itself. So to begin to answer this question, we need to first figure out if a human can get close enough to a neutron star to ever see one. As other answers have pointed out, new neutron stars are WAY to hot and radioactive to approach, but if you were to investigate an older, colder neutron star, you just need to get close enough to survive the gravitational sheer (assuming you have a spaceship up to the task.)
When you orbit an object you become "weightless" right? But that gets a lot more complicated with super dense bodies like a neutron star. Let's say you want to get close enough that the neutron star looks about the same size as the sun or moon, you'd need to orbit it at a distance of ~1150 km. At this range, you'd be orbiting it at a speed of ~12.7 million m/sec meaning you'd be spinning around the neutron star at a rate of 1.76 orbits per second. So, even if you could "see" the neutron star at this range, it would just be a blur to your pathetic human eyes. But the bigger problem here is gravitational sheer. Even if you got your speed and distance just right, the gravitational difference between the mass in your head and in your feet would be ~490 newtons per kilogram of body mass meaning that even if you could establish a stable orbit, your physical body could not survive.
So how close can you get?
The human body can only survive about 1.5 G forces for an extended period of time. Yes I know much higher is possible, but only for a few seconds, and it's often accompanied by visual distortion as your eyes compress, so for purposes of this question, I'll aim for the sustainable G force, not the extreme. Following this logic, it serves to reason that 14.7 newtons per kilogram of gravitational sheer is probably the most a human researcher could reasonably endure while he eyeballs the star. To get to this level of sheer, you need to orbit the neutron star at a distance of ~6640km. This will reduce your speed to ~5.3 million m/sec and an orbital period of about 7.8 seconds. At this distance, you'd be orbiting the neutron star slowly enough to "make out the surface" if it were old and cool enough, and the gravitational sheer would be low enough to conceivably survive.
Conclusion, what would it look like?
At this range, the neutron star would take up about 1/6th the size of Earth's moon in your sky. While other answers have talked about how cold a neutron star quickly becomes, even the coldest neutron stars are still hotter than the surfaces of most other stars, they are just too small to see with an Earth based telescope, but even a "cold" neutron star will still emit black body radiation; so, at this range, and older neutron star will likely look like a tiny blue sun. It may or may not be a bit too bright to stare at directly depending on exactly how old it is, but if you were to photograph it or look at it through a filter it would look a lot like any other star of similar temperature, but with some noticeable gravitational lensing around it.
Here is a picture of the moon placed next to a 10,000K neutron star at the closest observable distance for an idea of scale.
Searching for "neutron star appearance" you would have found this very similar question, from where I quote the answers:
A neutron star is very small for an astronomical object. 15–20 km diameter, this is like a medium size asteroid. From a distance larger than 50 thousand kilometres (in astronomical terms nothing, 1/8 Earth-Moon distance) it would be visible as a point. The problem is, that a neutron star is also incredibly hot, so if you approach it at that distance, you’ll be dead from radiation, cooked and evaporated before you’ve had a chance to see anything.
You can of course try to look from a larger distance using a telescope, but still, you’d need to be at a distance comparable to Earth-Sun distance, otherwise even a powerful telescope would not show you much more than a point. Also, you’d have to look at it through a strong filter, because high temperature means that neutron stars are very luminous. The surface of a neutron star is thousands times brighter than the surface of the Sun.
Therefore take all those “artist impressions” seen on the internet with a big grain of salt. It is not what it would look like from close distances. We won’t see neutron stars as anything beyond a brightly shining point for a long time ahead.
In the current 13+ billion year age of the universe neutron stars are all assumed to still be (...) well over 100.000 Celsius. That means looking at the currently existing population of neutron stars, they may be small, but that single pixel (tens of kilomters are hundreds of millions of kilometers) is bright enough to indude blindness many AU distant. If the neutron star is actively accreting matter, this will be much brighter. Note that this brightness will cook away matter in the immediate area, so most neutron stars won't have much of an accretion disk. Some may have "consolidated" moons composed mostly of iron or heavier materials. Anywhere else just cooks off into interstellar space.