What would it look like for a ship with an Alcubierre drive to drop out of FTL both for the passengers, and to an observer (assuming they aren't obliterated)?
First of all, I'm not a scientist of any kind so below is just my best guess. Hope it helps.
I would imagine a passenger would see nothing but blackness or possibly a faint diffuse glow while traveling, as the craft would be zooming by stars and planets faster than their light could reach a passengers eyes. The light which occupied the space along the skin of the warp bubble at any time would be squashed and smushed and unfocused to the passengers eyes leading to a faint low intensity glow.
(Or perhaps the light waves inside the warp bubble would bounce around the interior of the bubble endlessly, completely obfuscating the outside universe?)
When coming out of warp that light would begin to focus into points whose intensity would grow until the image resolved itself into the stars and worlds around the stopping point. (of course this would probably happen in a fraction of a second so might be hard to see with the naked eye)
From The point of view of an outside observer I would imagine some kind of flash or some other wave of energy to emanate from the point in space now occupied by the stopped craft as it would need to displace the energy or matter which occupied that space the moment before the craft appeared. The atoms occupying the space around the craft would be compressed massively as the craft appeared. Millions of atoms suddenly and violently moved in a uniform outward direction would be seen as an energy wave emanating from the stopping point.
This might not cause visible light to be emanated but some kind of detectable signature wave would appear to an outside observer with our current technology.
Assuming by dropping out of warp, you mean a gradual stop, you can find your answer here: https://worldbuilding.stackexchange.com/a/146605/57832. However, if you mean a sudden stop it will be similar with the exception being that you do not see the approach of a blue orb of light as it decelerates. Instead the blackhole in front of your ship will violently explode in a powerful blast of gamma rays and the ship will just appear in front of you and all the other stuff will happen the same from there.
From inside of the ship, instantly dropping out of warp will look the same as it does to the external observer. Because you will not experience your deceleration, you will be in the other person's reference frame to see the same black hole explosion and trailing ring behind your ship
Another scenario worth exploring is what a gradual deceleration would look like from inside of your ship. While you are moving >1C the bow of your ship will have a black blob in front of it, and the rear will have a mirror looking blob behind it. These are your black and white holes. Also, space itself will appear red or blue shifted around you at various angles depending on how fast you are going.
As you slow down below 1C, the white hole will start to look more plastic like, transition to transparent, then fade away. The black-hole in front of you will begin to glow as it starts bleeding off collected mass. While this glow will look blue shifted to the external observer, it will not be shifted at all from your reference point so the glow will be more white.
Also as you slow down the red and blue shifts around you will begin readjusting from being all red with a blue ring to just red in back and blue in front as you cross the 1C barrier. Then the shifts will fade away as you approach non-relativistic speeds.
You will also experience a Luminal Boom when you hit 1C. Just like air compresses into a sonic boom when you go the exact speed of sound, light will compress into very high energy gamma rays at 1C; so, for the health of your crew, you don't want to linger at 1C for very long.
I seem to recall calculations by physicists that showed a ship using an Alcubierre drive might potentially accumulate enormousness amounts of energy in its 'bubble' as it travels through space. This energy would then have to be released upon arrival at your destination as the warp effect is collapsed/turned off. This, they claimed could with disastrous effects for anyone (read entire planets/solar systems) in the vicinity such was the amount stored energy they claimed would be involved.
Given this is all hypothetical with widely varying calculations about things like for instance how much 'negative energy' you'd need to generate a working drive in the first place you are entirely free to reduce the amount of energy released upon arrival to something more realistic/survivable. The result would be a visible release of energy across a wide frequency range in front (or around) the ship advertising its arrival to outside observers.
Or you could potentially just add a requirement that your ship has to do short 'jumps' along the way to its destination in order to 'dump' the energy it's accumulating. This could be because some the energy 'leaks' inside the bubble as it accumulates and if you don't stop you fry your ship or alternately you simply want to make sure you arrive at your destination with a safe/manageable amount of energy stored in the 'warp bubble' so you don't kill anyone.
One note of caution - the only problem with this last approach is that you will turn any starship equipped with the drive into a civilization destroying super weapon. Just program an automated ship to take a long cruise without stopping so it arrives in your enemy's system with a huge amount of stored energy. Boom!