The authors generalize the results of Feigenbaum on the universal structure of period-doubling from real mappings to the complex case. They conjecture, that the fixed points of period n-tupling operators associated with infinite sequences of m/n period n-tuplings of complex mappings with quadratic critical points are universal and that the corresponding renormalization operators have a single unstable eigenvalue. The scaling numbers so obtained can be organized by Farey- trees and have well defined limits, which can be used to interprete the classical Feigenbaum number. The work mostly consists of numerical experiments and heuristic arguments and ends with notes on possible physical applications.