We can also define a multiplication operator which is commutative but not associative (same as the Mandelbulb multiplication operator):

I'm having trouble picturing how this is different from the "cosine method" (formula #3) on your summary page (http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/?action=dlattach;attach=527;image), with the x-axis here playing the role of the z-axis there. The formulas do look different I think, but I can't picture what the difference is geometrically. Aren't they both using spherical coordinates, but with the elevation angles measured as angle away from a pole rather than angle away from the equator as in the usual formula? In the picture for formula #3, I can almost imagine a Christmas tree pointing downward, maybe.

Edit: Try starting with formula #3 from the summary page and make the replacements , and I think it becomes the formula given on this page.