What does it mean to perform calculus upon functions of complex values? Complex numbers exist in a plane. This would lead me to believe that calculus views them as multivariate, but I am not real sure. How would one define a rate of change for a complex number valued function, or the area underneath it. Could someone explain this in simple terms?

At a very simple level, any complex number $z$ can be expressed as $x + i y.$
and $f(z) = u(x,y) + i v (x,y)$
The definition of derivative is the same definition.
$f'(z) = \lim_\limits{z\to z_0} \frac {f(z) - f(z_0)}{z-z_0}$
$\frac {\partial f}{\partial x} = \frac {\partial u}{\partial x} + i \frac{\partial u}{\partial y}\\
\frac {\partial f}{\partial y} = - \frac {\partial v}{\partial y} + i\frac {\partial v}{\partial x} $
if $\frac {\partial f}{\partial x} = \frac {\partial f}{\partial y}$ then the function is said to be "holomorphic."
