Why there isn't 3D 1:2 conformal transformation?

[Roquen]

Facepalm. I see my logic error. In case anyone in the future persists to attempt to poke a hole, let me attempt an explanation.  Sticking to R3, using my proposed construction the resulting space is identical R3 so 'f' is restricted to Liouville's.  Attempting to define a conformal "except" some subspace the same holds.  Take "except some plane", then the subspace on the positive side excluding the plane is conformal only of 'f' is Liouville's.  The same holds for the negative side.  Shrink the plane to a line then the two sides except the line must have the same 'f' to be conformal except the line.  The same holds for any choice of "except" subspace(s).  So any analytic function 'f' is either Liouville's or nowhere conformal.

[Alef]

Well, then no grails so far. Throught probably using alredy known transformations in more exotic spaces like hyperbolic space could result in something new. Or maybe not.

Comment claims ";This produces slice of the second order Mandelbulb directly perpendicular to the regular Mandelbrot set." Simple  cutout  of sine pow2 mandelbulb in y=0 plane:
z=z^2+c
z= abs(real(z))- 1i*(imag(z))

Cosine pow 2 mandelbulb in y=0 cutout looks lika mandelbrot set with slightly different reflection.
So mysterious mandelbulb too are demistified, mundelbulb formula just involves reflection.

However "The Rudy Rucker's formula for 3D Mandelbrot of 1991 year", a "Ruckerbulb" in perpendicular cutout generates something more like real mandelbrot, but then it have a lot of noise.

http://www.fractalforums.com/new-theories-and-research/complex-not-so-complex/msg61882/#msg61882