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As you know, when you square a complex number, you are squaring the absolute value, but the doubling the angle. So you can separate the square function into steps: separate the number into absolute value and argument, square the absolute value and double the argument, and recombine the number. Of course this can be generalized to any real power.

Well, you don't have to use the same power for the absolute part and for the angular part. For the mapping function, z=f(z), use:
f(z) = |z|^p1 * exp(i*arg(z)*p2)
Obviously, if p1=p2, we get the ordinary Mandel power. More interesting is if p1 != p2. (We don't have to limit p1,p2 to real numbers.)

What I found is that p1 controls how "plump" the interior is and the amount of chaos, and p2 controls the radial symmetry. If p1 < p2, the fractal looks thinner and more chaotic than the corresponding z->z^p2 fractal, and if p1 > p2, it looks obese and smoother. The minibrots look more distorted than normal. The imaginary part of p1 adds a swirl effect to the fractal.

I've plotted the p1=2; p2=3 case.

This is a pretty smart idea. I think I've seen this before, but I never saw it presented this way.