For better visualizing and understanding fractals like the Mandelbrot set, the idea of color cycling is a great invention.

Points outside the fractal are colored according to the number of iterations when a threshold assuring divergence ("bail out") is reached. Imagining the fractal bearing en electrical charge or a temperature, the points of same color, i.e. of same rate of divergence, form "equipotential lines" around it. Of course, those lines become more and more intricate as one comes close to the fractal.
So far, this is only static, but now cycling in time through the colors of the (periodic) color palette, either towards the fractal or outward, reveals so much more about its hard-to-see structures. E.g. for the Mandelbrot set, knowing that it is simply connected, cycling helps particularly in regions with spiral-like patterns to get an idea "where it is connected".
Just google for the terms fractal color cycling and you'll find tons of more or less hallucinating videos.