First example:

Image[
 With[{s = N[2 Pi], r = N[Range[-4, 4, 8/299]]}, 
  Map[{Arg[#]/s, 
     1, (# Conjugate[#])^0.33 &[Times @@ Sin[s ReIm[#]]]} &, 
   Mod[#/LogGamma[Abs[#]^-10.5] (1 + I Sin[s/2 Abs[#]]), 2] &[
    Outer[Complex, r, r]], {2}]], ColorSpace -> "HSB"]

Here's the result:

Second example:

Image[
 With[{s = N[2 Pi], r = N[Range[4/3, -4/3, -8/1497]]}, 
  Map[{Arg[#]/s, 
     1, (# Conjugate[#])^0.33 &[Times @@ Sin[s ReIm[#]]]} &, 
   Mod[Log[(1 + Abs[#])/(1 - Abs[#])]^0.25 # (-0.5 + I 0.1 Abs[#]), 
      2] &[Outer[Complex, r, r]], {2}]], ColorSpace -> "HSB"]

Here's the result:

Sugesstion for Mathematica 13:

ComplexPlot could have the option ColorFunction -> "CyclicReIm" additionaly to the existing ColorFunction -> "CyclicReImLogAbs", then these above functions could be expressed directly in ComplexPlot.

Some more functions generated with the code above (coffee beans?):

somehow hypnotic:

and even more whisked:

and simple again: