Double projection for 3d-mandelbrot

[Krumel]

So, here's a new idea for creating a 3d-mandelbrot:
   First, I'm quite sure, there can not be a real triplex-algebra.
   So instead I thought how you could do a 3d-mandelbrot with quaternions.
   Thus the proplem's mainly projection.
   Then I had this idea: How do you "make" a 3d-object to a 2d-representation?
      Right, you project it (with raytracing or whatsoever) on a 2d-plane.
   So why not projecting the 4d-mandelbrot on a 3d-cube, with the same method and then project this projection again?
I've not tried it yet, because it's horribly complicated and horribly CPU-intensive (I would be deeply grateful if someone would do it )