Pauldebrot, I know that the solutions of the derivative of the fractal formula give the critical points that can be used as the initial values of the Mandelbrot algorithm, however, I always got stuck when the result includes the parameter for the location in the complex plane.
For convenience I'll call the location in the complex plane c so the roots of the quadratic part become:
root 1 = 1 + sqrt((c-1)/c)
root 2 = 1 - sqrt((c-1)/c)
So for a root to be 0 then (c - 1)/c must equal 1 which also leads to a root of 2. Checking the pictures for initial values of 0 and 2 show that the first contains well formed Mandelbrot islands and the second does not, this indicates that the root 0 is indeed a critical point but 2 is not, currently I don't see why zero is a critical value.
Determining critical values for a fractal formula can lead to a wide variety of results, I've done this for cubics see
http://element90.wordpress.com/?s=cubic+observations and
http://element90.wordpress.com/2012/11/25/more-cubic-observations/, so determining critical values from root formulae that contain c values would be useful. Depending on the complexity of derivative of the fractal formula finding some roots is easy but others require much more effort.
Going back to the formulae for root 1 and 2, c is the location in the complex plane so values of root 1 and 2 are different for every location calculated by the Mandelbrot algorithm.
Here is a picture of the fractal when using zero as the initial value:
https://copy.com/TJxJSRTxekUgI have a generalised form of the fractal formula used for the pictures that started this thread:
z = (alpha(beta*z^gamma + delta*z^epsilon) + zeta)^eta
as it has multiple powers it is relatively slow.
So using this formula as a base I contrived a formula with i as a critical value:
z = (c(3z + z^3) + 1)^2
which produces:
https://copy.com/TBhgmUJnBPoEUsing other parameters for the generalised formula I came across a strange phenomenon of an unexpected need to use multi-precision calculation, see
http://www.fractalforums.com/general-discussion/unexpected-need-for-multi-precision/.
That's it for now.