What is a Basic Function? Surely z2+c the Mandelbrot Set is a basic function.
What about z + 1/z + c? In my opinion the basic function for the Nova Set, with scaling of the basic function (scalefactor * basic function).
In the table scaled 0.8 is the scale value of the function: scale*f(z,c): 0.8*f(z,c).
In the table s=0.25 is the start value 0.25 of the iteration. Please see math basics.
In the table the link "scaling" is a link to show the scaled variants of f(z,c).
Clicking on the thumbnail image, a large version is shown.

mabrot

scaled 1.0
s=0.0
scaling
nova

scaled 0.9
s=1.0
scaling
amoeba

scaled -0.8
s=1.0
scaling
blueAmoeba

scaled 0.7
s=1.0
scaling
bluedouble

scaled 1.5
s=0.0
scaling
bigfoot

scaled 1.4
s=0.0
scaling
\(\mathbf{\bf\:z^2 + c}\)
\(\mathbf{\bf\:z + \cfrac{1}{z} + c}\)
\(\mathbf{\bf\:z + \cfrac{1}{z} + c}\)
\(\mathbf{\bf\:\cfrac{1}{z} - z + c}\)
\(\mathbf{\bf\:\sqrt{z^2 + 1} + c}\)
\(\mathbf{\bf\:\sqrt{1 - z^2} + c}\)
disc

scaled 1.1
s=0.249962
scaling
shield

scaled 3.8
s=0.0625
scaling
moon

scaled 1.0
s=1.0
scaling
stone

scaled 5.3
s=1.0
scaling
novaTri

scaled 1.0
s=0.793701
scaling
brownNova

scaled 0.85
s=0.432041
scaling
\(\mathbf{\bf\:z - \sqrt{z} + c}\)
\(\mathbf{\bf\:\sqrt{z} - \sqrt{\sqrt{z}} + c}\)
\(\mathbf{\bf\:-z + \cfrac{1}{z} + \cfrac{1}{z^2} + c}\)
\(\mathbf{\bf\:\cfrac{\sqrt{z^2 + 1}}{\sqrt{z}} + c}\)
\(\mathbf{\bf\:\cfrac{\sqrt{z^3 + 1}}{\sqrt{z}} + c}\)
\(\mathbf{\bf\:z + \cfrac{z + 1}{\sqrt{z}} + c}\)
hedgehog

scaled 1.0
s=0.08830199
scaling
twoPole

scaled 0.9
s=0.37151
scaling
xNova

scaled 1.0
s=1.0
scaling
doubleMabrot

scaled 1.0
s=1.0
scaling
fergusonNova

scaled 1.0
s=0.295436901
scaling
expRow

scaled 1.0
s=0.0
scaling
\(\mathbf{\bf\:z + \cfrac{1}{\sqrt{z} + 1} + c}\)
\(\mathbf{\bf\:z + \cfrac{1}{z^2 - 1} + c}\)
\(\mathbf{\bf\:\cfrac{\sqrt{z^4 + 1}}{z} + c}\)
\(\mathbf{\bf\:\cfrac{(z + 1)^2(z - 1)^2}{2.5z} + c}\)
\(\mathbf{\bf\:z + 0.1*(z^2 - \cfrac{1}{z^2}) + c}\)
\(\mathbf{\bf\:e^{-z} + z + c}\)