Certainly: [itex]C^n[/itex] is the set of n tuples of complex numbers with the norm defined by [itex]\sqrt{\sum_{i=1}^n z_nz_n^*}[/itex] where the "*" indicates the complex conjugate.
But unless you have some kind of additional structure the uses the algebraic properties of complex numbers, that is identical to [itex]R^{2n}[/itex].