Due to a previous question being too sloppily written by me I here try to be a bit more clear on what I wondered.
For functions defined $$\mathbb C^{n\times n} \to \mathbb C^{n\times n} : A+Bi \to f(A+Bi)$$ for example by power series expansion: $$f(A+Bi) = \sum_{k=-\infty}^\infty C_k(A+Bi)^k$$ Can we derive results similar to those in analysis of one complex variable? Or more specifically for what kind of definition of analyticity and integral would it even make sense ?
What you are looking for is called holomorphic functional calculus. You can start from the content of that link, or books on Functional Analysis like Yosida's "Functional Analysis", Dunford and Schwartz "Linear operators". Or more basis ones like Conway's Functional Analysis should have a little of it.