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http://en.wikipedia.org/wiki/Wirtinger_derivatives

I would agree that this is not implemented in Sage but I would disagree that it can be defined as a "simple combination of the usual derivatives". In particular it is necessary to consider the chain-rule. I have a prototype implementation in FriCAS (Axiom) that does this. In that context I am interested in collaboration and critical review.

http://axiom-wiki.newsynthesis.org/SandBoxWirtinger

From the point of view of complex analysis you might also want to look up references to polygenic functions, e.g. Kasner

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076732/