There are two methods for defining a complex function in Mathematica.

Exploration.

Before we start, make x, y, u and v real variables.

Method 1.  Make  f[x,y]  a function of two real variables (x,y).

Method 2.  Make  F[z]  a function of  .  

Remark.  It was necessary to use the power of  MapAll  to find the real and imaginary parts of  .  

(c) 2006 John H. Mathews, Russell W. Howell