Sorry, but i had to do this. Some players fail to understand easy little facts. The link between Rapture and Columbia. It's easy, there isn't one. Those two places are completly different cities. As i'm not the author, nor director of Bioshock series, i can only guess they are in the same universe.
But there is link betwen Bioshock and Infinite. Bio in the name means something biological, in this case, something in human body. Like biological experiments prehaps ? Modifying DNA with ADAM and EVE ? Shock in the name means something sudden. As i mentioned before im not director of the series so i can only guess, but it could stand for sudden biological progress, or sudden fall of man made Utopia.
Let's have look at Infinite, shall we. Many players are asking, why call it bioshock if ADAM and EVE are missing. But are they really ? What do you think Vigors are ? A simple Coca Cola drinks ? They are, again, a DNA modificators. But this time, they're not persistant.
The main thing about Bioshock series is man made Utopia which fails. Why is that ? Easy. Which is good for one, may not be good for others. Which is utopia for some, may be hell for others. I think Columbia will be suitable for this "utopian dream" as much as Rapture was.
And finally, don't take Bioshock as some brand for underwater creepy city. Take it as some set of rules. The place and time can change, but the rules still apply (genetic modifications, failed utopia, dramatic storyline etc.). For instance, look at Final Fantasy series, different world with each sequel, but world rules stays same.
Final verdict. Columbia may not be an underwater city, but again. do you really want another Bioshock from Rapture ? I say, let's see what Columbia has to offer first, then we can jump to conclusions.