TITLE: Is Complex Analysis equivalent Real Analysis with $f:\mathbb R^2 \to \mathbb R^2$?
QUESTION [58 upvotes]: Am I correct in noticing that Complex Analysis seems to be a synonym for analysis of functions $\mathbb R^2 \to \mathbb R^2$? 
If this is the case, surely all the results from complex analysis carry over to the study of these $\mathbb R^2 \to \mathbb R^2$ functions. Does anything from complex analysis carry over into the study of functions in even higher dimensions? 
Furthermore, is there an area of Mathematics similar to complex analysis that investigates functions, say, $\mathbb R^3 \to \mathbb R^3$ to the same level of detail?

REPLY [9 votes]: The simple answer is that the complex analysis is over the complex plane $\mathbb{C}$, which is NOT equivalent to the Euclidean plane $\mathbb{R}^2$. The complex plane $\mathbb{C}$ processes additional structure defined by complex multiplication, which is not defined over $\mathbb{R}^2$.