In general the prequantum hilbert space of some manifold $M$ will be the sections of the complex line bundle. I ommitted a lot of details of course. Locally, a section of a complex line bundle simply looks like a complex valued function. (of course the sections are square integrable).
However, for a euclidean space, we choose $L^{2}(R^{2n})$ as the prequantum hilbert space. But aren't these real square integrable functions? Why not complex any more.