What is a Basic ZC Function? Surely z2+c the Mandelbrot Set is a basic function.
The standard way to have a fractal generation function is: z = f(z,c) = 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 "rendering" is a link to show different mathematical renderings of f(z,c).
Clicking on the thumbnail image, a large version is shown.

zcMabrot

scaled 1.0
s=0.0
rendering
zcMabrotc

scaled 1.0
s=0.0
rendering
zcMabrot2c

scaled 1.0
s=0.0
rendering
zcNova

scaled 1.0
s=2.0
rendering
zcNovac

scaled 1.0
s=-1.0
rendering
zcCanyon

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

scaled 1.0
s=2.0
rendering
zcSandDisc

scaled 1.0
s=2.0
rendering
zcTriple

scaled 1.0
s=1.88988157
rendering
zcLogisticMap

scaled 1.0
s=0.5
rendering
zcWhiteBars

scaled 1.0
s=2.46264186
rendering
zcCircleTriple

scaled 1.0
s=1.889881575
rendering
\(\mathbf{\bf\:\cfrac{z^2}{c^2} + \cfrac{c^2}{z^2}}\)
\(\mathbf{\bf\:\cfrac{z^3}{c^3} + \cfrac{c^3}{z^3}}\)
\(\mathbf{\bf\:\cfrac{z^3c^3 + 1}{zc}}\)
\(\mathbf{\bf\:cz(1-z)}\)
\(\mathbf{\bf\:z^3c^3 + zc + \cfrac{1}{zc} }\)
\(\mathbf{\bf\:\cfrac{z^3c^3 + 1}{z^2c^2} }\)
zcQuadruple

scaled 1.0
s=2.0
rendering
zcDumbbell

scaled 1.0
s=-0.300283106
rendering
zcDum

scaled 1.0
s=4.0
rendering
zcNovaInside

scaled 1.0
s=0.0
rendering
zcOutside

scaled 1.0
s=0.4
rendering
zcFerguson

scaled 1.0
s=-1.472470394
rendering
\(\mathbf{\bf\:\cfrac{z^4c^4 + 1}{z^2c^2} }\)
\(\mathbf{\bf\:zc - \cfrac{2zc}{z^2c^2 + 1} }\)
\(\mathbf{\bf\:(zc - \cfrac{1}{zc})^2 }\)
\(\mathbf{\bf\:\cfrac{z^2c^2 + zc - 1}{zc - 1} }\)
\(\mathbf{\bf\:zc + \cfrac{zc + 1}{zc} - 2.6 }\)
\(\mathbf{\bf\:z^4c^4 + zc - 1 }\)