Singular sets of harmonic functions in R^2 and their complexifications in C^2

It is well known that the critical set of (real) harmonic functions in B 1 ⊂ R 2 is isolated. In fact, the complex critical set of their holomorphic extensions is also isolated in C 2 . It is shown that the number of complex critical set in a fixed ball at 0 ∈ C 2 is bounded by a quadratic order of the frequency of harmonic functions in the (real) unit ball.