In his book Visual Complex Analysis (an awesome book, by the way), Needham, on the topic of graphing complex functions, says that
Actually, the situation is not quite as hopeless as it seems. First, note that although two-dimensional space is needed to draw the graph of a real function $f$, the graph itself is only a one-dimensional curve, meaning that only one real number (namely $x$) is needed to identify each point within it. Likewise, altough four-dimensional space is needed to draw the set of points with coordinates $(x,y,u,v)= (z, f(z))$, the graph itself is two-dimensional, meaning that only two real numbers (namely $x$ and $y$) are needed to identify each point within it. Thus, intrinsically, the graph of a complex function is merely a two-dimensional surface, and it is this susceptible to visualization in ordinary three-dimensional space.
Is this actually possible? I've been thinking for a while about how one would graph a complex function using only three dimensions, but I can't find a way.