IGN version has taken it Interview with the head of Take-Two Strauss Zelnik. He talked about the development of the expected game Bioshock 4.

Potion promises that the continuation of the famous series will definitely be released. According to him, although there is a problem in creating a game, it will become great. The authors want to maintain the heritage of the creator Bioshock Kena Levina is trying to meet the requirements of the audience.
Bioshock 4 will appear, I can say this on my heart, without any questions. We have ups and downs. We also had to change the studio leadership. However, we have a difficult task, with Ken Levin's heritage and the success of his games. We need to make sure that the new game deserves to be called Bioshock And it is a significant step compared to other parts. It seems to me that some opponents have realized too late that just normal games are no longer normal. Good – the new is bad, now everyone wants to excel. Our mission is to make a great game.
The previous information appeared that 2K games did not arrange a conspiracy Bioshock 4 and now the game will have to be redone. The project has been developed for more than 10 years, but some details about the plot of the new part are still unknown.