I have begun studying the theory of several complex variables and in particular, Riemann domains. We define a Riemann domain as follows.
A Riemann domain is a pair $(X,\pi)$, where $X$ is a topological space and $\pi : X \to \mathbb{C}^n$ is a local homeomorphism.
I am using two books at the present, Gunning and Rossi's Analytic functions of Several Complex Variables and Shabat's Introduction to Complex Analysis Vol II. Both discuss Riemann domains, but provide little to no examples of Riemann domains.
Of course, Riemann domains are the higher dimension generalisation of Riemann surfaces. Therefore, do Riemann domains arise if we consider the ranges of the multivalued function $f(z_1, z_2) = \sqrt{z_1 z_2}$, $f(z_1, z_2) = \sqrt{z_1} + \sqrt{z_2}$, $f(z_1, z_2) = \log(z_1 z_2)$, $f(z_1,z_2) = \log(z_1 + z_2)$?
Yes: see Fritzsche, Grauert, From Holomorphic Functions to Complex Manifolds, pp. 87-95.