You can vie view complex inner product space as an object that encompasses a set of real inner product spaces which are not necessarily positive definite. I.e. from $\mathbb{C}^n$ with the standard inner product you can get Euclidean real space as well as Lorentzian by choosing appropriate real form. So if you want to find a real geometric intuition for complex inner product spaces (assuming there is one!) you should probably better start with inner product spaces of indefinite signature.
As an example of use of inner product spaces I suggest the theory of simple Lie algebras. The theory is developed in the complex case and one has a natural (invariant) inner product there - the Killing form. From one complex Lie algebra you can get several real Lie algebras and their properties (i.e. whether they are compact or not) are determined precisely by the signature of the restriction of Killing form.
