Such a spacetime is called an ultrahyperbolic spacetime, so called because it produces ultrahyperbolic equations (equations with more than one negative eigenvalue). Those spacetimes are not overly nice to work with.
- They pretty trivially include closed timelike curves, since a closed curve in any plane of two time directions will be timelike.
- They permit particles to decay without any restriction regarding masses, including single photon decay. 
- A hypersurface of dimension $n-1$ is required to solve matter fields equations on a spacetime, but obviously such a hypersurface will not be purely spacelike in the case of an ultrahyperbolic spacetime. Hence you cannot predict events given informations on the current time.
There are still a few theories regarding it but overall it is not really that studied, for those reasons.