The key paper seems to be:
FRACTAL ASPECTS OF THE ITERATION OF z → Λz(1- z) FOR COMPLEX Λ AND z
Benoit B. Mandelbrot
Annals of the New York Academy of Sciences
Volume 357, Nonlinear Dynamics pages 249–259, December 1980
http://onlinelibrary.wiley.com/doi/10.1111/j.1749-6632.1980.tb29690.x/abstract(but full paper is pricy so I didn't read it)
So originally it was
, possibly named after the logistic map.
z is common in complex analysis, presumably because you need a variable name and
is the next in the alphabet, also common is
.
It turns out that all quadratic polynomials are equivalent to the z²+c form (conjugate is a keyword to search for), so the exact form doesn't matter so much when you get down to abstract mathematical properties.