>>"There seems to be something magical about complex numbers, but what?"

Complex numbers provide a concise way to describe *all smooth mappings that don't produce whipped cream*. Suppose we are looking at z->f(z)+c iterations (z->f(z,c) doesn't seem to add much). "Whipping" the cream means stretching some directions more than others. We can consider the Jacobian (J) at a point, i.e. d(z_new)/d(z_old), which measures the local amount of stretching/shearing/scaling/rotating at each point. For 2D this is a 2 by 2 matrix with the real and imaginary axes playing the role of the x and y axes.

What do these Jacobians look like? For z->z, the identity, J is the identity. From this we build up. For z->z*z, we multiply the complex plane by z. Following the rules of complex numbers, we get J=[2r -2i; 2i 2r], with r and i being the reals. This matrix applies a rotation and a scaling, *it has no shearing or stretching*. What about z->z*z*z? We apply a multiplication twice, which applies this rotation + scale twice; still no whipped cream. For polynomial functions the overall Jacobian adds the Jacobians from the linear (a simple scaling), quadratic, cubic, etc terms. Adding rotation + scaling matrixes is still a rotation + scale. The + c term simply adds the identity. As polynomials are universal function approximators, so long as the function is "analytic" (i.e. not piecewise defined) and doesn't use conjugates *any* function will be safe from whipped cream.

**The dose and the poison.** If the function isn't smooth everywhere, i.e. z^2.5, branch points will break up the fractal and any zoom of. If the function uses reflection, the C term will cause a minor whipped cream in places because an identity + *reflecting* rotation *will* distort angles, but for most iterations the +C term will add very little distortion. This opens up a lot of "shape weaving/stacking" possibilities such as the burning ship which can be deeply explored. There are many ways to piece together reflections of the complex plane through the origin making non-trivial deep-zoomable fractals. However, if the distortion is in the f(z) term we will get *distortion stacking* which builds up very quickly and makes deep zooms hard.

**Higher dimensions are barren.** Conformal mappings are only interesting in 2 dimensions. In 3 or more dimensions they are limited to Mobius transformations: rotation, translation, scaling and 1/z inversions. It's true that complex z*z only rotates, scales and translates on a *local* scale but it still is more "interesting" on a global scale. In 3 or more dimensions, such as the quaternions, even z*z causes, on a local scale, shearing/stretching and the whipped cream we see. Mandelbulb lives with whipped cream and the mandelbox combines a burningship-esque reflection trick with Mobius transformations. No shape weaving in the mandelbox, most "exploration" is shallow zooms as well as parameters, lighting, etc. So having multiple variables isn't helpful in the land of analytic function deep zooms, unless there is a clever slicing trick I am missing or something.

**Julia shape weaving plays with the root**. Passing near a minibrot to get an embedded Julia means passing near a root of quadratic or higher order. Roots crush the constellation of iterating points downward, allowing the +C term to matter as well as folding it up; both are very important for shape stacking. If we restrict to the smooth analytic functions without branch points, we have polynomials as well as essential singularities to play with. "But essential singularities aren't roots!". They still can have regions that act as roots, and if we pretend f(0)=0 we can get interesting pictures in some cases. Here is an example of shape weaving using an embedded Julia's edge as well as it's spiral for z->z*exp(1/z)+c: