The yet-to-be-named BioShock 4 is currently under development at Cloud Chamber, a studio with multiple locations established by publisher 2K specifically to work on the series. It would be too far of a stretch to call it the successor to Irrational, the Seattle studio led by Ken Levine that developed Bioshock and Bioshock Infinite, but Cloud Chamber does boast plenty of series veterans that have been involved from the start, including the game's design director Jonathan Pelling and a creative director with the best name in the industry: Hoagy de la Plante.
Cloud Chamber has now posted several dozen new job openings (first spotted by VGC) and says it "is currently recruiting like mad for game developers in two amazing locations" in San Francisco and Montreal. The 29 roles range across production, design and other skillsets, though notable by its absence is any mention of multiplayer (Bioshock 2 had a half-decent multiplayer mode, while Infinite was conceived of as a multiplayer title before abandoning the idea mid-development). There's not much known about Bioshock 4, but one of the sure bets is that it's a singleplayer shooter.
