Using the Chain Rule for Complex Functions

We just started studying complex valued functions in my calculus class and we were going through the rules of taking derivatives and limits. My professor mentioned that you cannot use the chain rule for certain complex valued functions. After doing some homework problems I'm curious about what he meant. For example if I want to take the derivative of (x+i)^-1 can I set u=(x+i) and differentiate using the chain rule. It seems to work when I do it. If you can do this, when can't you use the chain rule. Can somewhere please explain this to me and maybe give some concrete examples???