In my current lecture I regularly encounter usage of the real part of, say, a scalar product of two vectors similar to angles in classical geometry. For example in Hilbert space theory:
Let $H$ be a Hilbert space, $C \subset H$ be convex and closed. For $x_0 \in H$, $x \in C$ is the best approximation of $x_0$ by $C$, iff $\forall y \in C : Re \langle x - x_0, y - x_0 \rangle \leq 0$
As much as it is intuitive (and the proof itself is no problem), I do not know how to interpret this. So the question (not necessarily connected to the above example theorem) is
Is there any useful interpretation to the real part of a scalar product in complex vector spaces?
if there is none, in best case the real part is just used for convinience, and the author of my book wants to grasp a more general concept. - in worst case, not such interpretation exists.