I guess no one wants to continue this?

Welcome to the forum. Take a look in the Mandelbulb forum - there are quite a few discussions there to find the real McCoy.

Aka, the Mandelisk

I guess this goes here, though the idea is based on the Mandelbulb method, just using different angles:

UF iteration code (magn precalculated):

            magn = sqrt(magn)^@mpwr
            th = @mpwr*atan2(zri)                       ;az
            ph = @mpwr*atan2(imag(zri) + flip(zj))  ;ax
            sz = sin(th)
            zri = magn*(cos(th) + flip(cos(ph)*sz)) + cri
            zj = cj - magn*sz*sin(ph)
            magn = |zri| + sqr(zj)

Reslts in these:

z^2+c:



z^8+c:

OK, maybe there's still something better to find, this version has "correct" cross-sections in both XZ and XY planes for even powers>=2 but the odd ones are slightly off.
Anyway here's the iteration loop calculation as programmed in UF (all variables real except zri and cri which are complex, magn precalculated on loop entry):

            magn = sqrt(magn)^@mpwr
            th = @mpwr*atan2(zri)                      ;az
            ph = @mpwr*atan2(imag(zri) + flip(zj)) ;ax
            r = @mpwr*atan2(real(zri) + flip(zj))    ;ay
            sx = sin(ph)
            sy = sin(r)
            cy = cos(r)
            sz = sin(th)
            cz = cos(th)
            zri = magn*(cz*cy - sx*sy*sz + flip(cos(ph)*sz)) + cri
            zj = magn*(sx*cy*sz + sy*cz) + cj
            magn = |zri| + sqr(zj)

Note that |zri|+sqr(zj) is the square of the magnitude.

Edit: Apologies I overestimated how good the 2D cross-sections where - I thought some deviations from the "correct" outlines where errors in my clipping routine but it turns out that the cross-sections are close to being correct, but slightly disturbed in one way or another

Here's how I came up with the "Mandelview":

The usual way of transforming a scene for viewing in 3D is using the rotation matrices for example:
 
| cz -sz  0 |
| sz  cz  0 |
|  0    0   1 |
 
and
 
|  cy   0   sy |
|   0    1   0  |
|  -sy  0   cy |
 
and
 
|  1   0     0  |
|  0  cx  -sx |
|  0   sx  cx |
 
The reversal of the sign of the sines in the Y angle rotation being the "correct" method.
Now we can transform space using a combination of these matrices in any order, we want the order where the z is first, so either z*y*x or z*x*y.
I tried z*x*y first, i.e.
 
| cz -sz  0 ||  1   0     0  | = | cz   -sz.cx  sz.sx |
| sz  cz  0 ||  0  cx  -sx |    | sz    cz.cx  -cz.sx |
|  0    0   1 ||  0   sx  cx |    | 0        sx       cx   |
 
| cz   -sz.cx   sz.sx ||  cy   0   sy |  = | cz.cy-sz.sx.sy   -sz.cx   cz.sy+sz.sx.cy |
| sz    cz.cx  -cz.sx ||   0    1   0  |
| 0        sx        cx   ||  -sy  0   cy |
 
For the equivalent of the "Mandelbulb" method we only need that first row assuming you multiply like this (as I was taught):
 
|x y z|| a b c | = | x.a+y.d+z.g  x.b+y.e+z.h  x.c+y.f+z.i |
         | d e f |
         | g h i |
 
Because our vector to be transformed is |1 0 0| (pr more accurately |magnitude 0 0 |
 
Anyway I tried: magn*(cz.cy-sz.sx.sy, -sz.cx, cz.sy+sz.sx.cy) which gave a combination of Mandelbrot and Mandelbar cross-sections if I remember correctly.
Now a Mandelbar is just a Mandelbrot with a sign change so I just played around with the signs of the terms with sines until both cross-sections were Mandelbrots and ended up with: magn*(cz.cy-sz.sx.sy, sz.cx, cz.sy+sz.sx.cy) which is totally incorrect if applying "normal" rules - hence my comment on the animation regarding a secondary "imaginary" level since the "incorrectness" involves a sign change.

Afterwards I tried the z*y*x order and got a modified version of that which produced the correct cross-sections for the z^2+c version but still had the problem with the odd powers and in addition only one of the cross-sections was correct for each of the even powers above 2.

Edit: I have since found that some of the cross-sections are only close to the "correct" versions - I thought the difference was a problem with my clipping

Another alternative method of getting a true 3D Mandelbrot I've been considering is a 3D version of trigonometry using an idea that I've never actually seen described or used but I think is sound (I know I'm not rigorous enough for a proof):

Consider a right-angled tetrahedron (which is an object that can by defined simply by (x,0,0),(0,y,0) and (0,0,z) it's fairly obvious that any two such objects are similar in the same way as triangles if the ratios of the *areas* of the sides stay unchanged.
Considering this I thought that maybe we could define a specific right-angled tetrahedron by any two known ratios of areas - for example if we define the areas of the sides as xy, xz and yz for the small sides and xyz as the base (=="hypoteneuse") then if we know xy/xyz and either xz/xyz or yz/xyz then we have defined a set of self-similar tetrahedrons.
Is this correct ? If so it probably gives an alternative method of defining rotation in 3D and squaring, multplying etc. once we have a version of Pythagoras for tetrahedrons relatng the areas.

Edit: OK I know - make that the ratios of the square roots of the areas

The "Mandelview" cross sections are not as close to being "correct" as I at first thought, I initially thought the difference was a problem with my clipping algorithm but have since found that some of the cross-section are only similar to the "correct" versions.