How is it better to encode a complex number into RGB pixel and vice versa?

Probably (logarithm of) an absolute value goes to brightness and an argument goes to hue.

Desaturated pixes should receive randomized argument in reverse transformation.

Something like:
0 -> (0,0,0)
1 -> (255,0,0)
-1 -> (0,255,255)
0.5 -> (128,0,0)
i -> (255,255,0)
-i -> (255,0,255)

(0,0,0) -> 0
(255,255,255) -> e^(i * random)
(128,128,128) -> 0.5 * e^(i *random)
(0,128,128) -> -0.5

Are there ready-made formulas for that?