This is an extended re-post of a question that I have asked on MSE not a long time ago. But anyway, it seems more appropriate for MO.
To begin with, in a real inner product space we have a geometric intuition for the inner product. In the finite dimensional case, the inner product of two vectors is the product of their lengths (norms) times the cosine of the angle between them. Reverse engineering this suggests that the purely algebraic properties of an abstract inner product give it the properties of the "length of the projection" (scaled somehow). In particular, we can think of two vectors with zero inner product as orthogonal geometrically (and even call them that way) and bring all the geometric notions related to orthogonality to the abstract, possibley infinite dimensional case (with the appropriate care and restrictions of course). My question is, what is the geometric or otherwise intuition behind the abstract notion of a complex inner product space?
Here are a few thoughts:
1) This is a good mathematical structure to model some physical phenomena. An (if not the) example is quantum mechanics. This is an interesting line of thought. One problem is that I don't know enough quantum mechanics to follow it more deeply. If it is possible to explain in an elementary as possible way, What does it mean that a state of a particle is an element of a complex Hilbert space I would very much want to hear about it. I would also like to hear about other, hopefully more elementary, phenomena modeled by complex inner spaces. In particular, ia there any such phenomenon modeled by a finite-dimensional complex inner product space?
2) This is a good mathematical tool for other mathematical theories. Perhaps, unitary representations of groups or of other algebraic structures. Again, I don't have enough background in representation theory myself. explicit examples of such utility are welcome.
3) it is a good tool to investigate real structures by complexification and exploit of the good properies of the complex field (such as being algebraically closed). There are a lot of such examples in linear algebra, such as the classification of orthogonal maps, but I haven't seen such examples in the context of inner product spaces.
I would like to stress that saying that this is somehow algebraically natural analogue of real inner product spaces and that it has a lot of nice properties is somehow not enough in my opinion. Also not very satisfying is saying that it has application in such and such very advanced theories without elaboration.