I understand what you're saying, My motivation is that I want to find the "fundamental nature" rather than artifacts or pretty pictures. The regular z^2.5+c fractal has obvious artifacts when you pass over the branch cut, and I hope that my technique is more "true". (These terms are all hard to define, but I think I mean the same thing you do.) I've explored different escape bounds, for instance, to make sure that isn't causing artifacts. (The bound causes huge artifacts with negative exponents, by the way.)
It bothers me too that my fractals are less Mandelbrot-ish. I suspect this is due to all the branches overlapping and obscuring the details. I'm currently working on a couple ways to separate out these branches. Unfortunately just picking a different branch doesn't help, since it just moves the discontinuities. One plan is to display the fractal on the Riemann surface in 3D to separate the branches. The other plan is to crawl along the edge so you can continuously move from branch to branch without jumps. Julia sets are also on my list of things to do if I ever have time.
If you do experiment with this, let me know what you find out.
It bothers me too that my fractals are less Mandelbrot-ish. I suspect this is due to all the branches overlapping and obscuring the details. I'm currently working on a couple ways to separate out these branches. Unfortunately just picking a different branch doesn't help, since it just moves the discontinuities. One plan is to display the fractal on the Riemann surface in 3D to separate the branches. The other plan is to crawl along the edge so you can continuously move from branch to branch without jumps. Julia sets are also on my list of things to do if I ever have time.
If you do experiment with this, let me know what you find out.