Interactive Mandelbrot zoomer without using MandelbrotSetPlot

I really enjoyed this question about an interactive Mandelbrot zoomer in Mathematica and especially bill s answer because of the simplicity of his code. I would like to extend the code bill s provided such that it doesn’t use MandelbrotSetPlot but keep the Manipulate so I can zoom around. That way one could experiment and easily change the iterated function to functions like $z^3+c$ for example and also rational functions.

Code bill s provided:

Manipulate[b = -Log[a]; 
MandelbrotSetPlot[{u[[1]] + u[[2]] I - b - b I, u[[1]] + u[[2]] I + b + b I}, 
MaxIterations -> 200], {{a, 0.50, "zoom"}, 0, 0.999}, {u, {-2, -1.3}, {0.6, 1.3}}]