I've been looking at escape time versions of simple IFS fractals. This should help us think about the more complicated ones and 3-D versions. No claim of originality.

Here's the code and then a few pictures:
 
 x=real(z);y=imag(z);
  x=abs(x);y=abs(y);
  x1=x;y1=y;
       
  //Shape to be carved out of square
       
 // if ((x1<0.5)&&(y1<0.5)) {x=100;} // Menger
  if ((y1>0.5)&&(x1>0.5)) {x=100;}  // Viscek Cross
 //  if ((x1>0.875)&&(y1>0.875)){x=100;} // Thicker cross
         
  //Bounding square
  if (x>1.5) {x=100;}
  if (y>1.5) {y=100;}
       
  //Shifting
  x1=floor(x+0.5); if (x1>1) {x1=1;}
  y1=floor(y+0.5); if (y1>1) {y1=1;}     
  x=x-x1;
  y=y-y1;       
      
  //Rotation
  x1=a11*x+a12*y;
  y1=a21*x+a22*y;
       
  //Scaling
  x=3*x1;y=3*y1;
   
  // For now, pixel = 0
  z=x+flip(y);
  z=z+pixel;

visual:

A = [ a11 a12
         a21 a22]

A is a matrix transformation. In order to be conformal, it has to be a rotation matrix times a scalar. In the 1st and 2nd pictures, A is the identity matrix. In the 3rd, it is a rotation of 45 degrees.

The pixel is zero in all three pictures. It's hard to find interesting patterns with pixel > 0.

-mile