Well, when it comes to such things the interpretation of the meaning of the vector is always linked to the basis of the space, since they define it knowing what your basis is dirrectly affects the kind of image will be most helpful to visualize what's happening in the system you're studying .For example, in most cases in QM the basis is the eigen-functions of the Hamiltonian operator, which provides us, at least in common cases, with a countable basis, and that for physical/statistical reasons are integrable and square intregrable, which are nice enough properties for them to have a Fourier transform/ Fourier series form. So in this case the complex components of the vector represent the amplitude of the wave an it's phase, so when you multiply this vector with a complex number what you're doing is altering the amplitude and the phase of each wavepacket that builds your quantum state. If your vector space is the spin of the electron a different image would be most helpful.

EDIT: in short what i tried to say is this, the basis will give you a hint on what the best visualization for your problems is.