A closer look at what happens near the edge of those weird regions in the last animation. In this one, the stationary roots are at i and -i and the third root moves from sqrt(3) to about 1.8364. I also colored the roots white instead of black to avoid confusion.
Here's an example of a region where none of the points converge to a root. The orbits inside these regions "get trapped" and they don't escape, which is why they don't find roots.
These are also familiar shapes so it's kind of funny they show up here.
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