The BioShock 4 release date might have been leaked by the recent Nvidia GeForce datamine.
According to WCCFtech, the leak, which Nvidia has acknowledged as containing "released and/or speculative" PC titles, features a listing for 'Bioshock 2022'. That could mean that the new BioShock game announced by 2K Games and Cloud Chamber Studio in December 2019 might be releasing next year. There's also a listing for something called 'Bioshock RTX remaster', but no further reference to a release date or any other details.
When it comes to the new game, information is still pretty scarce - all we really know is that 2K said it would be in development "for the next several years," and that the Cloud Chamber had been quietly working on the project for at least 18 months before the announcement - a Kotaku report from April 2018 mentioned a new BioShock title in the early stages of development.
In late 2019, late 2022 was still three years away, which could still fit under that "several years" banner. It's also possible that 2022 refers to the financial year, which would end in early 2023. Either way, with no word from 2K and disruption from the Covid pandemic causing a number of public projects to be delayed, there's no guarantee that we'll get to see - let alone play - anything BioShock related next year, even if that was the team's original target.
While the next BioShock game is a pretty sure bet to come to PC, many of the rest of the games mentioned in Nvidia's list are a little less likely, with ports of everything from Demon's Souls to Super Mario Bros. With that in mind, take this information with a heavy pinch of salt, at least until we get even the slightest teaser from 2K.