[imok20]

That's not quite true – as you zoom in, you're no longer simply taking Re(c) = x pixel on the screen and Im(c) = y pixel on the screen. The more you zoom in, you're moving in on the Re/Im axes, which means you're no longer looking at integer values for the computation. You start looking at floating point values for x (Re(c)) and y (Im(c)) when computing the sequence. Therefore, the computer begins to do more and more floating point computations on, ostensibly, arbitrarily accurate fp.

EDIT: Added "pixel on the screen".