Criteria for an analytic function when the real and imaginary parts are $n$ dimensional

Question

Say we have two functions $$ f,g:\mathbb{R}^n\rightarrow\mathbb{R} $$

and we define some complex function $\mathbb{R}^n\rightarrow\mathbb{C}$ as $$ h(\vec x) = f(\vec x) + ig(\vec x) $$

If $n=2$, we have an easy criteria to check if we have an analytic function in the Cauchy-Riemann equations. Is there a similar analog for $n>2$?