I like to find out whether such a feat could be realised at all due to the sheer astronomical amount of energy required. I'm developing a story about super advanced alien civilization demonstrating it's science prowess, could they do it without invoking magic? Are such feat always dangerous?
"Teleportation" requires scanning each particle, destroying it, and reproducing it in a new location. Each part has problems. I'm going to use a "small" galaxy cluster of "just" 100 galaxies to make this "easier". That means a mass of about 1014 or 2x1044 kg, a diameter of about 7x106 light years, and a volume of 1.4x1021 ly3. I'm also going to assume you can (somehow) get all the information you need from a distance as well as totally annihilate and recreate particles at will (assuming you have the energy).
Scan an entire galaxy cluster is time consuming to say the least. If you try to do it from a single point the information you receive from light for nearby matter will be current, and from distant matter will be millions of years old. This will create a very lopsided view of the cluster. You need a snapshot in time.
First, position a sensor at the center of the cluster (this will take at least 3.5x106 years if you're not already in the center) and scan for 3.5x106 years. After that time you'll finally have received light from the edges of the cluster which left 3.5x106 years ago when you started scanning. Now you have a complete map of the cluster as it appeared 3.5x106 years ago.
You might think "ah ah! I'll send out Von Neumann Probes traveling near the speed of light across the whole cluster to scan it faster!" The problem is it will take your probes, from the center, over 3.5x106 years to spread throughout the cluster and then another 3.5x106 years for the most distant ones to report back. But we're going to need those probes anyway for the next phase, so might as well send them.
Now you have to destroy every single particle in the galaxy cluster, probably simultaneously else things go whizzing off in odd trajectories. Good thing you positioned all those Von Neumann probes all over it! Let's just assume they can do it... somehow. Don't ask me how you destroy a black hole.
This is the tricky part because according to E=mc2 our "small" cluster has a total energy of, oh dear, 1.8x1063 J. To give you an idea, a supernova has an energy of 1044 and this is 19 orders of magnitude more powerful.
That's just its rest mass. There's all that momentum to deal with with everything rotating and spinning and revolving.
You'll also need to store all the quantum information about every particle. This information carries with it a significant amount of energy in and of itself, but I can't even begin to estimate what it would be.
Let's assume you can somehow capture and channel all that energy. We're going to need all of it, exactly all of it, for the next step.
This is the comparatively easy part.
While you're busy scanning and destroying the cluster in its original location you've been positioning another set of probes all over the volume of it's new location. These will recreate the cluster as it existed in it's last complete snapshot... somehow. I'm going to assume they can create matter from energy with all the quantum attributes correct.
You need to pick a place to put your new cluster, and travel to that location taking even more time. You'd probably pick a void to minimize the amount of existing matter you have to deal with. Voids aren't entirely devoid of matter, they're just 1/10th less dense than normal. That means plenty of stars and things need to be destroyed and something has to be done with all that energy. Perhaps it's stored to help with the creation process.
The destruction probes have beamed the energy they received from destroying the cluster, along with all the information about their quantum state, to their exact counter part in the new location. Each of these then uses that energy to create their designated particles simultaneously and in the exact quantum state... somehow.
And there you have it! Teleporting a galactic cluster in three easy steps.
Well, it would take an awfully long time. Not just the millions of years to scan the thing, but the likely far greater time to position your creation probes at the even more distant construction site.
Anything immediately caught in the galaxy cluster sized beam of energy and information transmitted from the old location to the new location would be annihilated. This would be a cylinder with a circular area 1.50x1014 ly2. This should be minimized because it would diminish the amount of energy available for creation, perhaps by beaming it to a transfer station which then forms it into a narrow beam which is received by another transfer station at the construction site which sends it on to the proper probe. Also it might be annoying for the inhabitants of that swath of space.
Regardless of how you transfer it, this enormous beam of energy is a problem not only for anyone who happens to be in the way, but because of the mass-energy equivalence it still has the gravitational force of the original cluster but now it's barreling through space at the speed of light. It will probably disrupt anything "near" its path, and by "near" I mean "within 100,000 light years" dragging it along. Fortunately they won't notice for 100,000 years and then only on a galactic structure level.
Any inhabitants of the construction site will have a very bad day as their entire existence is transformed to matter to make way for the cluster's new home. Hitchhiker's Guide To The Galaxy got this one right.
Neighboring clusters of the new site will find their gravitational situation suddenly and extremely altered. Not so much that an individual planet, solar system, or even galaxy would much notice, but the stellar evolution of the nearby volume of space will be altered. Not to mention the same for the original site now devoid of 2x1044 mass.
Hopefully the beam won't drag too much matter along with it. Would be a shame if it slammed into your cluster's new home, or their neighbors.
Can this feat be performed without invoking magic? Yes and no.
Sufficiently Advanced Technology is Indistinguishable from Magic:
If you showed a demonstration of Quantum Levitation to a medieval human, they would probably conclude that it was magic because they have no frame of reference for how such a thing could be possible within the realm of physical law.
Except the alien's feat from the question amplifies the problem incalculably: a medieval human is still human, and possessed of roughly equivalent intelligence to a modern human--lacking only education and experience.
Conversely, the aliens may be able to individually comprehend principles beyond even the cumulative capabilities of all humans throughout history.
So instantly moving an entire galaxy would appear to be magical to most modern humans. But, if the aliens wanted to demonstrate their technical prowess, rather than claim any kind of supernatural ability, then they've got two avenues:
They have power to spare: Moving that much matte would be incomprehensibly costly, especially if a "brute force" method is used (ie, scanning, destroying, and re-creating every particle in the entire galaxy). The aliens could explain that their energy generation technology has advanced to the point where virtually any task is possible.
Size matters not: Alternatively, the aliens could demonstrate that their understanding of physics is so great, that they can essentially "edit the source code" of reality. In such a situation, moving a galaxy would be as trivial as moving a blade of grass.
The crucial aspect of the demonstration: If the aliens moved the entire Milky Way so far away that Earth could see no stars outside the galaxy, they would have to omit one crucial thing: light.
If the aliens did the galactic equivalent of a "drag and drop" on the entire galaxy, and included all the light that was present at the time, then people on Earth wouldn't notice a difference for hundreds of thousands of years; they'd need to wait until light (or the lack thereof) from outside the galaxy reached Earth.
However, if the aliens filter out all light when they perform the teleportation (which would be doable if they used the "source code" method above), then they'd get one hell of a demonstration. From the perspective of people on Earth, the sky would go black (light generated from terrestrial sources would return too quickly for anyone to see the interruption)...and the Sun would appear after about five minutes, when its light reached Earth.
The night sky would be utterly devoid of stars for four years, until Alpha Centauri would "appear" as its light reached Earth. Over the years, more and more stars would appear to "wink on" as their light reached us.
That would probably be a pretty effective demonstration of technological might.