He will not get the bends because the bends are caused by gas bubbles that form when the pressure is reduced, not increased. Since the elevator is descending, the air pressure is increasing so the gas bubbles will not form. The water pressure on the diver will not increase as the elevator descends because the water is not contained.
The exception to this would be if the geyser were of an arbitrarily large width/diameter so that the water column was effectively contained at the center where the diver was, with a resulting increase of water pressure due to the water column above the diver. This would be essentially the same situation as a diver swimming down the center a massive swimming pool that has had it's containment walls removed. There would be a period of time where there would still be high water pressure at the center of the pool even though the water column was not contained. Even then, the diver would only suffer "the bends" if he were to exit the geyser at the bottom of the elevator shaft and the elevator trip duration was of sufficient length at a high enough pressure to cause his blood to become oversaturated with gasses.
Another way to look at this is to imagine a 100 ft tall tank with a door at the bottom. If the diver swims down to the bottom of the tank and then exits the door, could he get the bends? The answer to that is obvious because the water pressure at the bottom of the tank is sufficient to cause the oversaturation of nitrogen in the blood that leads to the bends once the pressure is no longer present.
In fact, the entire original question is this: Is the water pressure at the bottom of an uncontained column of water greater at the bottom than it is at the top? I do not believe that an uncontained water column small enough to fit inside an elevator would have enough of an increase in pressure to result in the bends for a person exiting that water column.
