The standard way to have a fractal generation function is: z = f(z) + c, just adding c.
What about f(z,c) = z*c + 1/z*c? Where z and c are intermixed with multiplication. Or what about f(z,c) = z^c or f(z,c) = c^z ?
In the table scaled 1.0 is the scale value of the function: scale*f(z,c): 1.0*f(z,c).
Scaling f(z,c) functions in general is not of much value.
In the table s=0.25 is the start value 0.25 of the iteration. Please see math basics.
In the table the link "powered" is a link to show the powered variants of f(z,c): f(z^2,c^3), f(z^n,c^m).
Clicking on the thumbnail image, a large version is shown.

zcMabrot

scaled 1.0
s=0.0
powered
zcMabrotc

scaled 1.0
s=0.0
powered
zcMabrotc2

scaled 1.0
s=0.0
powered
zcNova

scaled 1.0
s=2.0
powered
zcCircleDouble

scaled 1.0
s=3.330190676786
powered
zcCanyon

scaled 1.0
s=2.0
powered
\(\mathbf{\bf\:z^2c + 1}\)
\(\mathbf{\bf\:z^2c + c}\)
\(\mathbf{\bf\:z^2c^2 + c}\)
\(\mathbf{\bf\:zc + \cfrac{1}{zc}}\)
\(\mathbf{\bf\:zc + \cfrac{1}{zc-1} + \cfrac{1}{zc+1}}\)
\(\mathbf{\bf\:\cfrac{z}{c} + \cfrac{c}{z}}\)
zcSand

scaled 1.0
s=2.0
powered
zcLogisticMap

scaled 1.0
s=0.5
powered
zcBasicFerguson

scaled 1.0
s=0.148148148
powered
zcTriple

scaled 1.0
s=1.88988157
powered
zcCircleTriple

scaled 1.0
s=1.889881575
powered
zcQuadruple

scaled 1.0
s=2.0
powered
\(\mathbf{\bf\:\cfrac{z^2}{c^2} + \cfrac{c^2}{z^2}}\)
\(\mathbf{\bf\:cz(1-z)}\)
\(\mathbf{\bf\:z^3c^3 + z^2c^2}\)
\(\mathbf{\bf\:\cfrac{z^3c^3 + 1}{zc}}\)
\(\mathbf{\bf\:\cfrac{z^3c^3 + 1}{z^2c^2} }\)
\(\mathbf{\bf\:\cfrac{z^4c^4 + 1}{z^2c^2} }\)
zcDum

scaled 1.0
s=4.0
powered
zcDoubleDivergent

scaled 1.0
s=-4.0
powered
zcDoubleDivergentLens

scaled 1.0
s=-4.828427125
powered
zcDoubleLens

scaled 1.0
s=1.0
powered
zcTriHead

scaled 1.0
s=-0.472470394
powered
zcDumbbell

scaled 1.0
s=0.300283106
powered
\(\mathbf{\bf\:(zc - \cfrac{1}{zc})^2 }\)
\(\mathbf{\bf\: \cfrac{ \cfrac{z^2}{c^2} } { \cfrac{z}{c} + 1 } }\)
\(\mathbf{\bf\: \cfrac{ \cfrac{z^2}{c^2} + 1 } { \cfrac{z}{c} + 1 } }\)
\(\mathbf{\bf\: \cfrac{ z^2c^2 - 1 } { z^2c^2 + 1 } }\)
\(\mathbf{\bf\:z^4c^4 + zc}\)
\(\mathbf{\bf\: \cfrac{ zc(z^2c^2 - 1) } { z^2c^2 + 1 } }\)