Complex valued functions of a real variable are common and important but well covered in real analysis. Functions of a complex variable I've never seen in practice.

A good course in statistics, e.g., with sufficient statistics, needs graduate probability, and with a grad probability course can do elementary statistics easily in the footnotes. E.g., the proof I worked out for the Neyman-Pearson lemma is quite general but based on the Hahn decomposition based on the Radon-Nikodym theorem (Rudin gives von Neumann's cute proof) in measure theory. The grown up approach to conditional probability is based just on the Radon-Nikodym theorem of measure theory.