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Recently Karl implemented a few functions which I suggested.
All of them are based on this function:

F(z)=z^2+z^n+c

Where n is any positive number.

The general shape of these sets is different but the shape of the minibrots is just like the mandebrot set (or higher power versions if the z^2 is replaced with a different power).

Does anyone know why this is?

https://imgur.com/a/Ah7u1

I tried some higher powers for the z^n. A new mandelbrot set grows out of it if you increase the power. And embedded juliasets can be found within the main set. These are also like normal mandelbrot sets.

Does Karl's implementation use the correct critical values?  The critical values for are which has 0 as an obvious solution but also n-2 other complex solutions .  I'm not sure what the correct way to render Mandelbrot sets with multiple critical values is - the decision criterion is either all escape for exterior or none escape for interior, not sure which is right.

For the 0 value, the periodic cycle of a minibrot will hit 0 again.  Close to 0, z^2 is much larger than z^n, so its folding-in-two effect will dominate - at other points in the periodic cycle the relevant region is small and far from 0 so the effect of the iteration is mostly linear with a small non-linear warping.  So I think the lowest power > 1 for any combination of powers decides what shape the minibrots are likely to have.