In April 2018 a report confirmed that the next BioShock game was in development within a "top secret" studio under publisher 2K. Since then, little has been said about this prospective BioShock game, which is understandable given the game's development was only just beginning. Yet BioShock fans have been hungry for more information about the game, especially given BioShock creator Ken Levine shifting to smaller, narrative-driven games. Luckily, some interesting information may have just been found.
A 2K job listing for an End Game Design Lead was discovered online, referencing a new team at 2K. The listing doesn't reference BioShock specifically, so it should be made clear that this is a rumor at best and stems entirely from speculation. The listing discusses the game being made at the studio vaguely. It references a "Fantastic World" and "Incredible Story," but nothing more. In other words, all that's tying this rumor to BioShock is that it's 2K hiring for an unknown team, working on a game with a story. It's admittedly thin.
That said, there's more to the listing that makes it much more interesting. Specifically, the job is for a senior designer who will help create a live service style of endgame content. The listing says that the game's story will end, but "The world, by contrast, is something we are designing to be alive and engaging past the main narrative arc." The job's tasks include designing quests, progression systems, and narrative structures to retain players post-launch.
In summary, 2K has a relatively small and new team working on a game that's planned to have a live service of some kind. The game will have a traditional main narrative arc, followed by this live-service endgame, not unlike say Borderlands 3, Diablo 3, or even the upcoming Marvel's Avengers. It's unclear if this game is the previously reported BioShock title, but that it's early in development and is coming from a new team within 2K does fit. But only just.
Again, it must be restated that this is a complete rumor. Live service games at 2K aren't exactly a rarity right now. Even the existence of a new BioShock project in development at 2K is a rumor, as it could have been a mistaken report or the project might have been canceled. If it still exists, it's very much still in early development, meaning even if this rumor is accurate the game could be canceled before ever coming to light. And even if it's true and continues in development, it will be years yet before any official announcement is made. Nevertheless, any BioShock news is interesting news.
BioShock 4 is rumored to be in development for unspecified platforms.