Falling elevator filled with water

During impact, there would be a sudden increase in the effective weight of the water above the man. This would cause a sudden increase in the water pressure which would collapse the lungs and probably drive air into the bloodstream. There would also be an increase in the pressure gradient. This would cause a sudden increase the buoyant force, which would be like landing on a hard surface, and would be likely to cause a fatal concussion.

I don't think this will be particularly different from the usual case of an empty elevator in free fall. When the elevator hits the bottom of the shaft, the elevator and everything inside must come to a stop over the course of a fraction of a second. Whether or not the elevator is filled with water, the person inside will experience sudden and dramatic acceleration when it hits the floor (i.e. they will go splat). From a drop of 20m, the person is traveling at 72kph when they hit the floor, and 0kph very shortly thereafter. That's going to hurt no matter what, regardless of if the person is stopped by the elevator floor or a layer of water.

Even if there are buoyant forces due to pressure gradients when the box of water hits the floor, anything that moves the person toward the ceiling of the elevator will only increase the acceleration they're experiencing, making the collision even more deadly. You might get some interesting physiological effects from the pressure shockwave traveling upward through the water, but there's nothing here that makes the situation any more survivable than your typical falling elevator.

To use this superpower to survive, the hero needs to fill the elevator shaft at the very beginning of the fall, and empty it slowly to gently bring the floating elevator to the bottom. Stopping suddenly at the bottom is going to be bad news no matter what.

The advantage compared to falling to the hard ground is that the stopping is not sudden, and $F_t = frac{dp}{dt}$ what means a smaller total force.

Once hitting the ground, the water buoyancy force acts on the human body at once $F_b = mu Vg$. But there is also the drag force. It depends on the final velocity and also on the position of the man. It is like the impact of diving after jumping from a great height.

It helps if the man stays vertical, if he stays close to the roof, and if the elevator cabin is tall. Otherwise it is like jumping in a shallow swimming pool.

I think it is tempting to assume that the sudden deceleration at the bottom will cause damage regardless of the presence or absence of the water. I will argue against that assumption.

Modeling your body as a bag of water, and supposing the elevator cab does not rupture, there is no tendency for it to deform during impact--nothing physically sets it apart from the rest of the water, so it doesn't get slammed against the bottom or unduly stretched or compressed along any axis.

During the impact, the pressure increases rapidly. The high pressure isn't necessarily an issue--the human body can survive pressures of tens of atmospheres--but the suddenness of the increase is going to cause problems. You can say goodbye to your eardrums, and the abrupt collapse of your lungs won't feel good. It would be akin to being in the water near an exploding depth charge.

In short, you might die, but your corpse will look much more presentable than it would without the water.

If you want empirical verification of this, I suggest filling a Nalgene bottle with water, inserting an egg, and dropping it from increasing heights. I think you may find you can drop it out of a second story window without breaking the egg.

The main damage here will be caused by the incompressibility of water. (Actually, the relative incompressibility of the water compared to the human body.) The effect would be somewhat similar to a human body entering water at a high speed - the incompressibility of water means it is like concrete.

In the case of the falling water-filled elevator, a shock wave would hit the human body from all directions and compress it. Since the human body has parts of various compressibilities (lungs are very compressible, fat and muscles are somewhat compressible, bones less and teeth quite incompressible), the soft organs would likely be very damaged: they take the brunt of the compression.