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Given that the wavefunction has to be a two-component object, why
is it so convenient to use complex numbers to encode the two
components? Beats me. Always seemed a little spooky, to tell
the truth.
Why spooky? There are lots of cases where we do similar things, e.g.
bundling 3 or 4 scalar equations into one vector equation. Whenever there
are N solutions to the same equation, it is just a matter of time before
somebody contrives a formalism that allows us to bundle them together.