In 2012 I found a fractal by using a fundamentally different approach. It arises when you colorize the complex plane by giving each pixel a grey value that corresponds to the percentage of gaussian integers that it can divide:
There are surely infinitely many more ways to generate other families of fractals though
I don't doubt there could be an iterative formula that maps to it, but I'd be very surprised.
It's also unclear to me that every iterative f(z,c) formula will produce something visually interesting, or indeed that meets the definition of "fractal".
Complex numbers have two components. If both are integers, the complex number is a Gaussian integer.