The intel comes via Kotaku, where reporter Jason Schreier uncovered the development of a successor to Bioshock Infinite while researching an article about the development of Mafia III and studio Hangar 13. "They're careful about not falling into the same problem every studio has, where they have too many people and nothing for them to do." explains one person now working on the new BioShock - at once revealing that they're still early in development, but that there's a clear plan in place for a core construct to build the title around. However, considering the game's story it's hard to see where the franchise will go.
However, the issue is that Parkside is not ready to bring a lot of Hangar 13 people over to work on the upcoming BioShock. It seems like it was a source of tension within Hangar 13, everyone competing to get a chance to work on BioShock.
The new rumblings on the franchise stem from a a Kotaku report on Mafia III, entitled "How The Makers Of Mafia III Lost Their Way". "They're careful about not falling into the same problem every studio has, where they have too many people and nothing for them to do".
It's not a surprise that a new BioShock game is in development considering the popularity of the series. Of note, Take-Two's November 2017 fiscal report teased that a "highly anticipated new title" in "one of 2K's biggest franchises" will be released on March 31, 2019. It's certainly possible that he was talking about BioShock 4, but we won't know for sure until the notoriously tight-lipped publisher makes it official.