BioShock 4 is coming eventually. Publisher 2K confirmed it was in development back in 2019, but said little more than that it would be several years away. The lead creatives are known: design director Jonathan Pelling, creative director Hoagy de la Plante, and art director Scott Sinclair, all of whom have extensive experience with the series all the way back to the original game.
Now Youtuber Colin Moriarty has provided the first substantial grist to what will doubtless become an ongoing rumour mill. Speaking on his podcast Sacred Symbols, Moriarty says: "It takes place in a 1960s Antarctic city called Borealis. [The game's] codenamed 'Parkside'… I’ve been told that the development team has incredible latitude to get it right."
The 'Parkside' codename has come up before: Specifically, in this 2018 report about Hangar 13 and this 2019 report about a cancelled version of Bioshock 4. Isn't the whole point of a codename that nobody knows it?
"Internally the game is very secret and apparently, totally locked up," added Moriarty. "Apparently the inclination there is that they understand full well that this game will be compared to what Ken Levine does. And by the way, [2K] is also publishing Levine’s next game."
As with any rumour, take all of this with a huge pinch of salt (except the bit about Levine's next game, which is obvious—his studio Ghost Story Games is also part of Take-Two Interactive). If you want to believe, then both Eurogamer and VGC says this lines up with what they've heard. 1960s Antartica immediately has me thinking of The Thing (incidentally, this time period would place it as happening around the same time as Bioshock 1 & 2).
Bioshock 4, which other rumours suggest may be called Bioshock: Isolation, is in development at Cloud Chamber Games, and has been the subject of many more rumours than I'm going to reproduce here. Some can be more-or-less confirmed thanks to job listings—the game has been said to be open world, for example, and Cloud Chamber has advertised for writers "who can weave impactful, character-driven stories in an open world setting."