Adding some imagination...

[David Makin]

Here are two renders of the Mandelbulb extended to complex "power" - (6,1) in this case.
I checked the XY slices and they unfortunately do not match the complex z^(6,1) + c Mandelbrot - the "bottom" looks correct but the bulbs are missing off the top - looking at the renders it looks like they're displaced in z on the Mandelbulb.

In the formulas note that ztemp, zri, cri and @power are complex and the rest are real.
Also |zri| is x^2+y^2 where zri = (x+i*y).

"-sine version":

            if (r = |zri|)>0.0
              ztemp = ((r=sqrt(r)) + flip(zj))^@power
              zri = real(ztemp)*(zri/r)^@power + cri
              zj = cj - imag(ztemp)
            else
              zri = cri
              zj = cj - imag(flip(zj)^@power)
            endif




"+sine" version

            if (r = |zri|)>0.0
              ztemp = ((r=sqrt(r)) + flip(zj))^@power
              zri = real(ztemp)*(zri/r)^@power + cri
              zj = cj + imag(ztemp)
            else
              zri = cri
              zj = cj + imag(flip(zj)^@power)
            endif

[David Makin]

Regarding:

            if (r = |zri|)>0.0
              ztemp = ((r=sqrt(r)) + flip(zj))^@power
              zri = real(ztemp)*(zri/r)^@power + cri
              zj = cj - imag(ztemp)
            else
              zri = cri
              zj = cj - imag(flip(zj)^@power)
            endif

I've assumed the zri/r is zero if |zri| is zero, would it be better to assume it is (1,0) instead ?
I did try it in the remote chance it would fix the XY cross-sections but it made no difference at all to the renders.