I don't think it makes sense to say you "lose" something. Yes, the properties of different spaces _are_ different, but these differences are intentional so that

I don't think it makes sense to say you "lose" something. Yes, the properties of different spaces _are_ different, but these differences are intentional so that they can be used to study something useful. For instance, complex analytic functions have a number of properties that don't hold for real analytic functions, so you actually "gain" something. An example is the residue theorem. One interesting thing about the residue theorem is that you can use it to calculate the integral of a real-valued function that wouldn't be calculable using conventional calculus techniques. So not only do complex functions have a number of interesting properties on their own, they can also be used to expand what's possible in real functions.