I feel like there’s a separate thing hiding in plain sight here which makes the Wirtinger calculus confusing: an entanglement with the algebra of complex numbers (complex multiplication and conjugation). I’ve felt in the past that this somewhat defies a clear geometric interpretation.
To take one term from the linear approximation formula…
\frac{\partial f}{\partial z} \cdot (z - z_0)
there, I’ve added that offending \cdot in explicitly.
While the Wirtinger derivatives are exactly “just directional derivatives” of a 2-vector valued function, their interaction with the complex delta z-z_0 and its conjugate always seemed essentially algebraic to me.
Does anyone have a clear geometric interpretation they would like to share?