In this Wikipedia article It is written the following:
A hyperplane in a Euclidean space separates that space into two half spaces.
Is this precise? I mean a hyperplane in $\mathbb C^n$ does not separate $\mathbb C^n$ into two half spaces. Indeed, the complement of a hyperplane in $\mathbb C^n$ is connected, it is even path connected.
This is a notational issue in a sense. What people usually mean by Euclidean space is something of the form $\mathbb{R}^n$. While it is true that $\mathbb{C}^n$ viewed as a vector space over $\mathbb{R}$ is isomorphic to $\mathbb{R}^{2n}$, we never call $n$-dimensional complex space a Euclidean space. The property you mentioned holds true for real $n$-dimensional space.