[LONG POST]

Finally I got it. The "triplex numbers" are really quaternions in disguise, and the "z = z^N+c" is really "v = |v|^N (q(v)^N i q(v)^-N) + c", where v is a 3D vector and q(v) is a function from a 3D vector v to a quaternion. For example, in the reversed-sign definition (the one giving (x,y,z)^0=(1,0,0), in the "old" ;-) terminology) you have these:

1) rotation by theta along z is expressed by cos(theta/2) + k sin(theta/2)

2) rotation by -phi along y is expressed by cos(phi/2) - j sin(phi/2)

and multiplying them gives

q(v) = cos(theta(v)/2) cos(phi(v)/2) + i sin(phi(v)/2) sin(theta(v)/2) - j cos(theta(v)/2) sin(phi(v)/2) + k sin(theta(v)/2) cos(phi(v)/2)

q(v)^-1 = cos(theta(v)/2) cos(phi(v)/2) - i sin(phi(v)/2) sin(theta(v)/2) + j cos(theta(v)/2) sin(phi(v)/2) - k sin(theta(v)/2) cos(phi(v)/2)

where theta(v) and phi(v) are the usual functions arctan(y/x) and arcsin(z/|v|) for conversion to spherical coordinates.

As a sanity check, let's see what happens on the xy plane. Remember that, at every iteration, a different rotation is applied!!! However since the elevation is 0 for the initial vector, they will all have phi(v)=0 and q(v) takes the form

q(v) = cos(arctan(y/x)/2) + k sin(arctan(y/x)/2)

then

v = i x + j y

q(v) i q(v)^-1 = i cos(theta) + j sin(theta)

|v|^2 q(v) i q(v)^-1 = |v|^2 (i cos(theta) + j sin(theta))

which is of course complex squaring, just changed to use i/j instead of real/imaginary.

This may well mean that this is not "the real 3D mandelbrot"; after all, why should one definition of q be better than the others? Or it may mean that they are all "real 3D mandelbrots" of course. It depends on how optimist you are.

So, the ultimate quest is to study the space of q(v)...

That's way beyond my knowledge of math, but there is a lot of other work to do; I am halfway through the math for the cartesian<->polar conversion, for example. Can you compute q(v) in cartesian coordinates? It is not the same (see the other thread) to compute q(v)^N using repeated multiplication of quaternions and to use repeated multiplication of triplex numbers. (The latter doesn't work). Why?

Triplex numbers still have their place in various ways, and they are worth further investigation. First because they are faster than quaternions :-) (or are they?...). Second because the quaternion notation for mandelbulbs does not extend to 4D or more, so a deeper understanding of the "triplex number" representation is fundamental to extend the mandelbulb beyond 3D.

Possible steps, beyonds those I have already outlined, include:

1) to try and understand what triplex multiplication actually means (if anything) and what it means that triplex numbers commute; maybe there is a better, noncommutative but associative definition of multiplication?

2) to convert my definition of triplex exponential in terms of quaternion exponentials and see what happens;

3) to try and understand what it means that powers of triplex numbers use a (-pi, pi) range for the elevation (see the other thread). How does this relate to cartesian formulas for triplex exponentiation? What is different between computing v^n and v*v*v on a triplex number?

So far, everything seems to make sense... It's kind of funny that the Mandelbulb started from "let's go away from quaternions", and ends up again dealing with quaternions!...