You don't need anything too complex** to do triplex "algebra".  You simply require:

A) a square root function to calculate a magnitude:
  r1= sqrt(y^2+z^2)

B) a complex power function:
  complex_1= (x + i r1)^n
  complex_2= (y + i z)^n

C) a real power function:
  r3=r1^-n       (you are applying the magnitude of y and z two times (once in each complex number), so need to divide it out once)

D) the ability to directly access the real and imaginary components of the complex numbers:
  new x = real part of complex 1   + x pixel value  OR  x Julia seed   (for Julias use the seed, Mandys use the pixel value)
  new y = imaginary part of complex_1 * real part of complex 2 * r3  + y pixel value  OR  y Julia seed
  new z = imaginary part of complex_1 * imaginary part of complex 2 * r3  + z pixel value  OR  z Julia seed

    Bugman has defined a triplex polar form using a particular matrix product.
    I have extended this idea to give 48 different polar forms.  If you are
    interested then please have a look at:
        http://soler7.com/Fractals/Matrices%20to%20Triplex.pdf

    Here is an image from one of the 48 variations:
        //soler7.com/Fractals/3D17/Mandelbulb1585.JPG
    Many more are at:  http://soler7.com/Fractals/3D0.html
    All were made using Visins of Chaos.