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.
You can vie 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.