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Someone's mentioned this already but I think for the 3D holy grail what we actually need is the X-Y and X-Z slices of the 3D z^2+c Mandy to be the 2D complex z^2+c Mandy and the Y-Z slice to be the 2D z^5+c Mandy

Yep - put together so that when X is zero we get z^5+c where z is made up of Y and Z and of course if Y or Z are zero then we get z^2+c where z is made up of X and Z or X and Y respectively.
Ideally this should be done such that unwanted terms are removed by cancellation rather than using the obvious scale by other dimension but I haven't looked into it and it's probably not possible to do that.

I got something, IMHO result of mixing powers 2 and 5 were pretty visual fractal. X value from xz plane unused, but it seems that stretchedness depends coefficient 0.
This was easy solution, mixed power bicomplex and far from holy grail. Maybe putting two x= y= z= polynomials together and moving around and changeing coeffittients of x could lead to something. Or maybe not;) But this stump have small buds and looking closer there are more buds and tinny stalks, feature expected but lacking in many other versions, just stretched shape.

I'm also hunting for the Holy Grail. I got some new ideas (at least I hope they are new) ... which look promising. But I have a problem: I do prefer to program this idea (instead of feeding some third-party software with the formula and waiting for it to cook an image). And the problem is that I will get a collection of pairs [3D point]-[color]. I have no idea how to convert this cloud of points into a solid object or a surface or a mesh.

Maybe there is some simple but effective algorithm for this?

Ah, as for the formula -- I prefer to keep it secret until I see what image does it generate. To be honest, the probability to get something ugly and unholy is extremely high (>99%).

 I toyed with a similar idea a little bit ago (maybe this thread subconsciously prompted it??...). 

1.  I took the 2d xy Mandelbrot, did edge checks for each point on the x-axis 
      --- is point on the edge of the fractal and inside fractal?  if so....
      --- check for next point... if y>1.5 go to next x point

2.  used the y value of this point to divide the y[/b] and z[/b] plane magnitude, then calculated the z^5 (tried others as well) Mandelbrot escapes for the yz fractal. 

  You end up with a z^5 yz cross section that follows the z^2 Mandelbrot xy.  It's a pain in the ass to implement in ChaosPro, and I didn't bother streamlining/debugging the code as it wasn't all that pretty. 

  Basically you get a relatively smooth pattern of transitions- exactly what you would expect.

  The problem being, at least for ChaosPro, there isn't a really easy way to implement something that checks the x axis first, then the yz plane vs. points on that axis.  Instead, you check points in whatever order CP desires, gotta recalculate xy plane radius for every yz pixel... TAKES A LONG TIME!@#!@#

  You'll notice that I didn't debug the script- has completely filled in sections "outside" of the x- radius because I didn't bother writing in a part of the script for that part (which would bailout...).

  Here is a z^2  x  z^5, straight on... only 333x333 pixels because it takes forever to calculate on my old ass computer:



  Here is the inside of a z^2 by z^2... gives more of a feeling of whats going on:


  The reason it is jaggedy (has those sudden cut offs) is pretty simple: it's determined by the resolution I check the xy radius at.  That resolution is the thing that slows the whole thing down in ChaosPro.  Now, for a directly written routine?  It wouldn't be a problem.  The thing is, I can't do that in CP....  basically have to achieve bootstrap levitation in order to do any of this stuff.

   I've got a busy weekend coming up... but maybe I'll help out with a flowchart so someone can implement this formula in MB or some other thing.  You guys will figure it out.