This is advice to Rubrikon and a reply to Goodies's comment that:
+1 wonderful scenario but I'm glad there isn't any "reality check" involved. I think the two large moons would eventually stabilize their orbits in eternal opposition. I'm afraid your two large moons won't have a common full moon. That would mean they are on the same side of the planet. Considering the large relative weight of our moon, a second one, forming a binary moon system with an Earth-like planet, will probably stabilize with the moons in orbit on either side of the planet, with no common full moon.
That would not happen in real life.
There are many examples of multiple moon systems in our solar system. There are no examples of 2 moons orbiting at the same distance from the planet and always opposite to each other. And moons in different obits will always change their relative positions as they orbit.
Part One: Moons in different orbits.
If there are two moons, They will (almost always) orbit the planet at different distances. And so the moon with the inner orbit will orbit faster and pull ahead of the outer moon until it is 180 degrees ahead, on the opposite side of the planet, and then catch up with the outer moon until it eventually passes it, over and over again. Sometimes the two moons will appear close together, sometimes only one will be seen at a time.
Sometimes one moon will be seen in the day, and the other moon will be seen in the night. Sometimes both moons will be seen in the day. Sometimes both moons will be seen in the night, making the night purple as requested.
With different red and blue moons, there will be constant changes in the color of the night.
Part Two: Moons in Exchange Orbits.
It is commonly said that the moons Epimetheus and Janus of Saturn share an orbit, but that is not exactly true. They have very similar orbits, but one orbits slightly closer to Saturn than the other. Because of the very similar orbits and orbital speeds, it takes many orbits round the planet for the inner one to pull ahead of and then catch up with the outer one, and they orbit so close that every time the inner one catches up their gravitational interactions make them switch obits.
Epimetheus's orbit is co-orbital with that of Janus. Janus's mean orbital radius from Saturn is, as of 2006 (as shown by green color in the adjacent picture), only 50 km less than that of Epimetheus, a distance smaller than either moon's mean radius. In accordance with Kepler's laws of planetary motion, the closer orbit is completed more quickly. Because of the small difference it is completed in only about 30 seconds less. Each day, the inner moon is an additional 0.25° farther around Saturn than the outer moon. As the inner moon catches up to the outer moon, their mutual gravitational attraction increases the inner moon's momentum and decreases that of the outer moon. This added momentum means that the inner moon's distance from Saturn and orbital period are increased, and the outer moon's are decreased. The timing and magnitude of the momentum exchange is such that the moons effectively swap orbits, never approaching closer than about 10,000 km. At each encounter Janus's orbital radius changes by ~20 km and Epimetheus's by ~80 km: Janus's orbit is less affected because it is four times more massive than Epimetheus. The exchange takes place close to every four years; the last close approaches occurred in January 2006, 2010, 2014 and 2018. This is the only such orbital configuration of moons known in the Solar System (although, 3753 Cruithne is an asteroid which is co-orbital with Earth).
If your planet had two moons in very similar orbits like Epimetheus and Janus, they would be close together in the sky for a long time, and be farther away in the sky for a long time, and would be far enough apart that only one was visible at a time for a long time. Years could pass when only one moon was visible at a time, when the red and blue moon alternated in visibility and when only one lights the night sky at a time. And then after that there could be years when the two moons were close enough that they usually were seen together and light the night sky together.
If you want such a sequence of lighting effects you should go with a Janus and Epimetheus type orbit. And naturally some characters would worry that maybe something will go wrong the next time the moons exchange orbits, and fear that one of the moons might crash onto the planet.
Part Three: Moons in a Trojan Orbit.
With 2 moons in a trojan type orbit, they will both be the same distance from the planet, and one will alwys be about 60 degrees from the other. Being separated that far will mean that sometimes they will be seen together and sometimes one will be below the horizon while the other is visible (and sometimes both will be below the horizon). More importantly, one of the moons would have to be tiny compared to the other for long term orbital stability, and so it will reflect only a tiny amount of the light that the other one does.
Part Four: A Ring of Moons.
You could have a ring of 7 to 42 moons of equal mass equally spaced in the same orbit and alternating red and blue in color of surface material.
But if the moons were massive enough to reflect enough light, orbital stability would probably require that the planet be too massive to be habitable for humans.
Maybe the habitable world would be a giant moon orbiting around a giant planet, and the ring of alternately red and blue moons would orbit the planet outside the orbit of the habitable moon.
Part Five: Double Moon - Two Moons Orbiting Each Other as They Orbit the Planet.
In a comment Ilmari Karonen suggested a double moon, with the two moons orbiting each other as they orbited the planet.
That might be the best way to have moons visble together for the mos time. They would be two equal size moons orbiting around their center of gravity or barycenter as they orbit the planet. The orbital distance between the two moons would be a rather small fraction of the total circumference of their orbit around the planet. Both the moons would be visible almost half the time. Both the moons would be out of sight below the horizon almost half the time. The red moon would be seen alone low above the horizon a small percentage of the time, and The blue moon would be seen alone low above the horizon a small percentage of the time.
They don't have to be exactly the same size. If one had half the diameter of the other one it would have a quarter of the surface area to reflect light, and one eighth of the mass of the larger one. It might appear to be only one quarter as bright as the other moon which may be a problem mixing colors to make purple light.
But if the two moons have different surface materials which have different colors, those surface materials could have different albedos, reflecting different percentages of the light that hits them. The larger moon's surface could refect a relatively small percentage of the light that hit it and the smaller moon could reflect a higher percentage of the light that hit it.
The variation in the albedo of various objects in the solar system is great enough that a moon with only one quarter the surface area of another one could reflect much more light than the larger moon.
And of course the star of the system might not emit equal amounts of red and blue wavelengths of light for the moons to reflect back at the planet. In fact it would probably be almost impossible for a star with a habitable planet to do so.
Of course designing a stable planet-moon system with a double moon orbiting the planet might be tricky, but it should be possible.