The residue field extension is not always trivial in characteristic $0$. For instance, in $\mathbb{A}^2_{\mathbb{R}}$, consider the plane curve $C$ of points $(x,y)$ satisfying the equation $x^2+y^2 = y^3$. The closed point $(0,0)$ of $C$ has residue field $\mathbb{R}$, yet the inverse image in the normalization is a single closed point with residue field $\mathbb{C}$.