This hypothetical generator is driven by the pressure differential between the inside and the outside.
Exactly.
That’s why I was wondering if it was possible. We’re getting rid of the pressure by using it run the turbines - essentially changing the pressure we don’t want into energy we do want. And I was thinking that maybe once we’ve converted the pressure into energy, there might be a low energy way to get rid of the now depressurized water.
I’m still curious whether the buried in mud solution would work.
ETA:
I have no idea on the technical details, but in principle wouldn’t that imply a perpetual motion machine?
This, however, is not going to appreciably reduce the pressure on the outside. It will, however, tremendously increase the pressure on the inside, until it equals the pressure on the outside.
I doubt it. As long as there still is a difference in pressure between the inside and the outside, you’ll have to overcome that in order to remove water from the inside.
Yes.
You could have a shaft running to the surface. Then you only have to overcome the gravity (which I figure ought to come out around the same since it’s gravity causing the pressure in the first place).
No. Getting buried under the mud would just add the pressure of the mud to the pressure of the water, not to mention the incredible difficulty of getting under the mud in the first place.
No, because it’s not an enclosed system. You’re receiving your energy from the pressure outside. It’s no more a perpetual motion machine than a windmill is.
You’d have to expend energy to remove water from that shaft. I assume that all the energy you’ll get back by refilling it through a turbine generator will not exceed that initial effort.
I’d probably try harnessing any underwater current. With 12,000 feet of water pressure behind it, even a crawling current speed would have some serious oomph behind it, I’d have to imagine.
Don’t forget that the current will also have 12,000 feet of water pressure in front of it.
How likely are the fauna (or even flora) to complicate things on the sea floor? I mean, it seems to me you’re just one distracted sperm whale away from having your day ruined.
Hehe oops 
People have already talked about using water to generate oxygen and filtering it for drinking/farming. Couldn’t you use this method to generate power and then go about your other uses for it since it’s already in?
Sure, until you run out of room. The water you brought in for drinking is eventually excreted by the occupants as urine; some day you’ll run out of room in the tanks of the porta-john’s, and you’ll have to start tossing milk jugs full of urine out of the window - and seeing as how the ocean around you is pressurized, you’ll have to do a considerable amount of mechanical work to get that stuff out there. In fact, at least as much work as you were able ot extract from the water rushing in in the first place.
Yeah but since it’s a current, that pressure is creating a vacuum in that direction.
If you’re extracting energy provided by the pressure of the water, then you need a low-pressure reservoir to dump the water to. That’s the interior of your habitat, and here we’re back to the original problem of limited space in there. If you’re thinking “aha, I’ll just dump the water to the surface of the ocean,” well, it takes energy to pump it up there; notice that a pipe from the surface of the ocean to any arbitrary depth does not magically flow water from that depth up to the surface.
If you’re extracting energy provided by the current (the kinetic energy of the water), then the static pressure is irrelevant; a current of any arbitrary speed provides the same potential for power generation whether it’s at the surface or at a depth of 12,000 feet.
Nobody’s thinking that if you have a shaft to the surface you can just run a wire down that to provide power and you don’t need a pressure driven turbine?
Nerds and tunnel (heh) vision. What are you going to do? 
Maybe we could have a miniature black hole to dump all this excess water into?
Well, how does the pressure of the water at 12,000 feet down compare with the load borne by a mine shaft under an equivalent distance of rock? It seems to me that it’s not only the pressure of the water that makes it dangerous, but also its extremely fluid state. I was wondering if really thick mud – either found on site or piped in from another location to cover the habitat – could mitigate that problem.
Not if your splitting the bulk of it into gases. Then you could store it in large pressurized tanks. You could probably get those up to the surface and sell them to fund the habitat. Heck you could even dump any excess gases into the oceans if you needed to. At 12,000’ the ambient pressure is apx. 5345psi. You can get tanks of 5000 easy, I’m sure that higher pressure ones could be gotten or made for a large operation like this.
The only question is how much energy a system like this would produce. I imagine it would run somewhat like a hydroelectric dam, anyone know how efficiently a small one might operate?