John Baez posted something on Google+: "This movie shows the sense in which Julia sets are self-similar"
It shows an animation that Anders Kaseorg made in answer to a Quora question about why Julia Sets are fractal.
Here's some more versions of the same idea:
The nice thing about this approach is that it explains how the Julia set works. Compare with the diagrams of the affine transforms in Iterated Function Systems. The key idea in both cases is that the fractal maps onto a smaller version of itself.
The z2 transform provides the 'stirring' motion but we don't normally see it. Even this (very good) video showing the orbits doesn't make this clear. Anders' trick for showing all the intermediate steps of a single iteration is wonderful.
The obvious thing is to make a similar animation for the Mandelbrot. As John points out, it's not as simple, because the set doesn't map directly onto itself every iteration. But it is still very pleasing!
Here's the same thing shown on a checkerboard instead of on the set itself. This makes it clearer that it's the transform that gives rise to the shape of the set.
Another way of animating this is to use a simple linear interpolation between each iteration. We lose the mathematical connection to z2 being a 'stirring' movement but it gives a simpler behavior:
This the closest I've come so far to the original goal of visualizing how the Mandelbrot works.
Compare with the images of integer iterations. They show the same thing but it's not obvious how to get from one frame to the next:
→ → ?