Hermitian Metric and Geodesics Why isn't general relativity developed with a Hermitian metric and a theory of complex valued paths and geodesics?
The concept of arc length and geodesic suffers under a pseudo-Riemannian metric. My readings of GR work around this by using different arc length definitions for time-like and space-like paths. I find this unsatisfying, especially when changing coordinates.   
 A: The theory of Hermitian manifolds only applies to even-dimensional Riemannian signature spaces. Many theories of gravity include higher dimensions, but even if we stick to the standard number of four dimensions, it's just a physical fact that we live in a Pseudo-Riemannian manifold. And the "un-satisfying" feature of different types of paths--timelike, spacelike, or null, is actually in my opinion a marvelous feature of the unification of geometry with physics. Beyond personal opinions however, the simple fact is that this is just the way the world is. All of special and general relativity rely on this structure.
That being said, the math of hermitian manifolds is very nice and pleasing, and it seems like nature often finds a use for pretty math. The theory of complex differential geometry is very important for the Euclidean path integral approach to quantum gravity, where many of the known instantons are hermitian manifolds. Also, these spaces play a very important role in compactifications of string theory which preserve supersymmetry. 
A: My interpretation of your question is: I accept special relativity as described on flat Minkowski space-time, now I wonder why we have to restrict ourselves to a real manifold in general relativity with a pseudo-Riemannian metric.
One of the main ideas behind general relativity is that, locally, it has to look like special relativity, which is the background of the laws of physics in an inertial frame of reference, i.e. free fall). Therefore this dictates that the manifolds one is entitled to consider in general relativity should be modelled on Minkowski space-time, which is a 4-dimensional smooth manifold equipped with a pseudo-metric.
In this regards, if you accept special relativity, most of the structure of general relativity comes for free, and there is not much space left to choice.
