(published on 05/21/09)
This is a nice follow-up.
The set of complex numbers is indeed a 2D vector space if you pay attention only to what happens when you add the numbers. However, what makes complex numbers special is their multiplication rule:
This is very different from the generic 2D vector dot product, ac+bd, which gives a number, not another vector. This complex multiplication rule makes the complex numbers form a mathematical field (http://en.wikipedia.org/wiki/Field_(mathematics)), something you can then multiply vectors by to get the same type of vector you started with. In other words, it lets complex numbers be used as scalars, just as you use real numbers. In physics we routinely use them as scalars, in that they don't change under spatial rotations, unlike say standard 3-vectors or 4-vectors.
You can measure things like ac magnetic susceptibility that we represent, with good reason, as complex numbers. It's not necessary to use that representation, but it's very convenient. I think what you're probably more interested in is the complex quantum wavefunction. With some awkwardness, it could be represented without explicit complex numbers, but whatever mathematical expression was used would end up being exactly equivalent. The wavefunction isn't quite measurable, since there's always an arbitrary absolute complex phase factor. (Relative phases are defined.)
The role of vector spaces in quantum mechanics goes far deeper than the mere use of complex numbers. The quantum states themselves form a (generally infinite-dimensional) vector space. Physical measurables, like energy, are represented by linear operators on this vector space, like matrices on finite-dimensional vector spaces. The time-dependence operator is length-preserving, so it's just a rotation in the abstract vector space. Once you start thinking of quantum mechanics in terms of these state vectors, you'll get hooked.
As for the philosophical questions about intrinsic reality etc., we have little to say.
(published on 06/07/13)