This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
Zero is the critical point of z = 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, the critical point is given by 2z + 1 = 0, so start with
z = -0.5. 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),
Note that you can precompute the first Mandelbrot iteration by
starting with z = c instead of z = 0, since 0^2 + c = c.