This article is from the Fractal FAQ, by Ermel Stepp firstname.lastname@example.org with numerous contributions by others.
Zero is the critical point of z^2+c, that is, a point where
d/dz (z^2+c) = 0. If you replace z^2+c with a different function, the
starting value will have to be modified. E.g. for z->z^2+z+c, the
critical point is given by 2z+1=0, so start with z=-1/2. In some cases,
there may be multiple critical values, so they all should be tested.
Critical points are important because by a result of Fatou: every attracting
cycle for a polynomial or rational function attracts at least one critical
point. Thus, testing the critical point shows if there is any stable
attractive cycle. See also:
1. M. Frame and J. Robertson, A Generalized Mandelbrot Set and the
Role of Critical Points, _Computers and Graphics_ 16, 1 (1992), pp. 35-40.
Note that you can precompute the first Mandelbrot iteration by starting with
z=c instead of z=0, since 0^2+c=c.