All the visualizations of the mandelbrot set I've seen use a threshold of iterations that define the edge. I think that threshold is more significant than any instability due to floating point numbers. A more formal analysis of numerical stability comes down to whether the iterated function is considered well-conditioned. I don't have the math to state it formally, but my intuition is that the mandelbrot function is, in that a small change to the function input will have a small effect on the number of iterations except at the very edge where the threshold will define the boundary anyway.

Yeah, I didn’t intend it as a formal proof, but it is an easy argument to make.

I’m sure an error limit can be given, as it uses a quite basic arithmetics only.