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It may well be this is just an alternative way of the standard Mandelbulb - but it doesn't look quite the same - here's a very quick test render (at *30,0000 when you see the whole thing at *1, solid at 1e-6 DE)

Althougn the first version I posted is "flatenned" I kinda like its spikiness so here's a render I did before "fixing" it by reducing the magnitude of z instead of just subtracting from it.

how sweet, a new candidate

right, please elaborate your idea, combining complex number has always been an attempt to create a 3d number system, what was your idea ?!

the z^2 variant is definately a lot bigger in the z-direction
and the z^8 variant has some whipped cream along the bulbs, but quite similar!

The idea came from thinking that if you have more dimensions than 2 for the vectors then the *angles* should be more than 1 dimensional - this struck a chord since the problem's always been that the 3 (or more) values concerned need changing simultaneously and single dimensional angles can't do that, even if you have more than one you always have the problem of order of application.
So I played with various ideas for complex angles - initially trying atan2(z)+flip(atan2(real(z)+flip(real(w))))) but that was interesting but not even close - in fact like just about nothing I've seen before
I tried that and several similar options because I was attempting to strictly stick to a pure 3D or triplex type system but after about 3 hours of trying I gave up and switched to 4D.

The method is straightforward - it's just that we use the angles such that the (complex) sine is z/magnitude and the (complex) cosine is w/magnitude (where z and w are complex giving 4 dimensions).

Then I just used the normal equations for multiple angles i.e. sin(2a)=2sin(a)cos(a) and cos(2a)=2*cos(a)^2-1 - hence for the original squared version:

  float m = |z|+|w|   ; magnitude squared
  if m>0                   ; we use a division so make sure we don't have zero
      w = 2.0*z*w      ; w = (new mag)*sin
      z = 2.0*sqr(z)    ; z = (new mag)*2*sqr(cos) : note no -1 as we need to apply along the z vector not just to the real part
      float n = cabs(z) ; magnitude of 2.0*sqr(z)
      z = z*(n-m)/n    ; equivalent to (new mag)*(2*sqr(cos)-1) if it was just real
  endif

After the above just add constants as normal.

Also multiplication and I think division are pretty straightforward and I think it's a field - but maybe someone else can work that out

One possible alternative that I can think of that in a way makes slightly more sense is if the magnitude is defined as either cabs(z)+flip(cabs(w)) or maybe cabs(real(z)+flip(real(w)))+flip(cabs(imag(z)+flip(imag(w)))).

One point to mention is that the object is truly 4D but we're just viewing in 3D - and I think there are some disconnected bits on the z^2+c version plus of course some "whipped cream" however I think this may be explained by the fact that the DE used is doing the calculations in 4D therefore it could and will detect points of close approach to "solid" outside the 3D space *and* because it's a 3D slice of a 4D object the disconnected bits may well be connected via the 4th dimension which of course is cut as far as the view is concerned.

The other weird thing is that for some reason I initially chose to use "2*cos(a)^2 - 1.0" for cos(2a) instead of cos(a)^2-sin(a)^2 and if you switch to using that instead.......

Nice. There are some streching on its back, but expected 3D mandelbrot features are not only on z=0 sides but alsou on y=0 top and bottom. And some strechings could be a feature of something I don't know, maybe low iteration number or DE;)

In picture 2 fractal looks as zoom on each stalk would look different. As on  my version holy of grail, which however hadn't nice stalks on the top. And there are thicker stalks on x>0, and much thinner objects on smaller bulb where x goes to -1 (?). Maybe try different zooms in different regions of z^2 as I did in that thread.

Interesting zoom in 3rd picture, most of holy grail versions haven't anything what would resemble 2D stalks. Throught these things on stalks looks as z^8 lakes and not as z^2 minibrots. Maybe cutting them at z=0 would reveal mandelbrot lakes or maybe it's the iteration number and DE. As when you zoom on stalk with low iteration number and see something round, then increase iteration number and mandelbrot appears.

Alsou woun't be bad to see a julia sets.

Anyway, nice insect:) The most grailish of them all.