Given the setting in BioShock, they probably developed a special volume or something similar to get the effect. I would suggest looking at DM-OceanRelic, but I think it's basically using a water texture on the "dome" above the playing area.
I think there are some "bubbles" in the game (emitter), so those can help. But the distortion effect might be the thing that sells the underwater look. While I don't know if it's possible, you might be able to achieve something similar with a post processing volume.
But I am guessing to really achieve it would take some programming, but who knows.