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 |
| cy 0 sy |
| 0 1 0 |
| -sy 0 cy |
| 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