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vec3 domainMap(vec3 c)
{
  float s = dot(c,c);
  return c/s;
}

z = domainMap(z);

// coordinate to invert to infinity
uniform vec3 InvertC; slider[(-5,-5,-5),(0,0,0),(5,5,5)]

// performs the active c = T(s)
vec3 domainMap(vec3 c)
{
  float s = dot(c,c);
  return c/s + InvertC;
}

z += domainMap(-z);

const float M_PI = 3.141592653589793238463;
const float M_PI_2 = (M_PI * 0.5);
const float M_PI_4 = (M_PI * 0.25);

// Map 'c' within unit square ([0,1],[0,1]) to a unit disk
vec2 toUnitDisk(vec2 c)
{
  c += c;

  float a = c.x - 1.0;
  float b = c.y - 1.0;
  float t,r;
  
  if (a*a > b*b) {
    r = a;
    t = M_PI_4*(b/a);
  } else {
    r = b;
    t = M_PI_2 - M_PI_4*(a/b);
  }
  
  return r*vec2(cos(t),sin(t));
}

vec2 zortho(vec2 a) { return vec2(-a.y, a.x); }

vec2 domainMap(vec2 c) { return toUnitDisk(zortho(c)); }

vec2 domainMap(vec2 c) { return toUnitDisk(c)-vec2(1.0,0.0); }

complexGrid = 
  Compile[{{max, _Real}, {n, _Integer}}, 
   Module[{r}, r = Range[-max, max, 2 max/(n - 1)];
    Outer[Plus, -I r, r]]];

complexHSB = 
  Compile[{{Z, _Complex, 2}}, 
   Module[{h, s, b, b2}, h = 0.5 + Arg[Z]/(2 Pi);
    s = Abs[Sin[2 Pi Abs[Z]]];
    b = Abs[Sin[2 Pi Im[Z]] Sin[2 Pi Re[Z]]]^0.25;
    b2 = 0.5 ((1 - s) + b + Sqrt[(1 - s - b)^2 + 0.01]);
    Transpose[{h, Sqrt[s], b2}, {3, 1, 2}]]];

domainImage[func_, max_, n_] := 
  ImageResize[
   ColorConvert[
    Image[complexHSB@func@complexGrid[max, 2 n], ColorSpace -> "HSB"],
     "RGB"], n];

domainPlot[func_, max_: Pi, n_: 500] := 
  ContourPlot[0, {x, -max, max}, {y, -max, max}, Contours -> {}, 
   RotateLabel -> False, 
   FrameLabel -> {"X", "Y", ToString@StandardForm@func@"z"}, 
   BaseStyle -> {FontFamily -> "Calibri", 14}, 
   Epilog -> 
    Inset[domainImage[func, max, n], {0, 0}, {Center, Center}, 
     2` max]];

domainPlot[# &] (* identity *)

F[z_] := Abs[Re[z]] + I Im[z]  (* the fold used in kaliset *)
domainPlot[F[#] &]

F[z_] = Sqrt[z*z]      (* note relation to the kaliset fold *)
domainPlot[F[#] &]

domainPlot[Sin]         (* zeroes along the x-axis yield one origin per zero, unless there's a constant added, in that case the constant will be the point of the base fractal mapped to the zero *)

 z += domainMap(pos-z);

if (i<ColorIterations) orbitTrap = min(orbitTrap, abs(vec4(domainMap(-z),r)));

uniform vec3 InvertC; slider[(-5,-5,-5),(0,0,0),(5,5,5)]

vec3 domainMap(vec3 c)
{
  float s = dot(c,c);
  return c=s * InvertC;
}

  if (i<ColorIterations) orbitTrap = min(orbitTrap, abs(vec4(domainMap(pos),r*r)));

vec2 domainMap(vec2 c)
{
  //c = 0.5*c;             // only effects resulting scale
  c = zpow2(c) - c;      // give the domain two zeros
  c   +=K ;
  return c;
}