How is the complexification of spacetime justified? As always the caveat is that I am a mathematician with very little knowledge of physics. I've started my quest for knowledge in this field, but am very very far from having a good grasp.
General relativity assumes that spacetime is a $4$-dimensional manifold. Ultimately, I know that physicists deal with Calabi-Yau manifolds (in string theory, I imagine). These are holomorphic complex manifolds.
How, and when, is this change of hypotheses justified? Is one way to view the general relativistic space time that spacetime is a variety of dimension $2$ over $\mathbb{C}$? Or is this change done later? Why is it ever done?
 A: The Calabi-Yau manifolds used as compact dimensions in string theory are ordinary real 6-dimensional (or real 8-dimensional, in the case of an F-theory description) manifolds and you don't have to distinguish them from other manifolds over real numbers.
However, they're special 6- or 8-dimensional manifolds that may also be described as 3-dimensional or 4-dimensional complex manifolds. That's because there are natural complex coordinates and many things are holomorphic etc. In particular, the monodromy group isn't the generic $SO(6)$ or $SO(8)$ but the subgroups of $U(3)$ or $U(4)$ which preserve the merging of the pairs of real dimensions into the complex ones.
Complex manifolds in this sense are just special examples of real manifolds of a doubled dimension. They're special examples with special properties - in particular, such complex manifolds preserve supersymmetry under some additional assumptions, and so on.
Independently of that, one may also complexify spacetime dimensions by analytically continuing in them. This is much more natural in the case of momentum spaces - and the continuation to complex values of energies and momenta is possible and useful because the scattering amplitudes are analytic functions of the momenta and energies.
The continuation to imaginary values of time is useful in thermodynamics - a thermal ensemble is obtained from a periodic Euclidean (imaginary) time - and has new special features in quantum gravity - the black hole solutions may be Euclideanized and these solutions nicely shrink and become regular near the event horizons (like a cigar).
So the question of complexification has many facets and it seems that you are a bit confused what you're exactly asking about. Different things have different justifications. But the very general idea of yours that one has to have "justification" is wrong. In physics, the justification is that something works. On the contrary, you have to have a justification if you claim that there is a problem with any of these methods and procedures. There is no problem with complexification of space, time, momentum space, or other things. Because you haven't really proposed any conceivable problem, it's not possible to answer your question in more detail. One can't be solving a problem if there's no problem to start with.
A: Only for BPS configurations in supersymmetric theories.
