Normally a Taylor series is constructed along real numbers. However, for practical purposes mathematics often heralds that commonly known continuous functions in the real plane are equivalent to their own Taylor series in the complex plane as well.
Suppose I want to construct a Taylor series for $f(x)=\sqrt{x}$ (the real variable), but, I want to expand about the complex number, say, $(1+i)^{2}$. Is the process for constructing the series about that complex number any different than constructing a Taylor series about a regular real number? Otherwise, what do I need to do differently?
The term you are looking for is analytic function. The procedure is no different for the complex case.