I know some facts about the geometry when coordinates are real. In pseudo-euclidean spaces (like in special relativity) T is also real, just the definitions of a distance is different.

R^2 = X^2+Y^2+Z^2 - T^2

But we can say that it is an euclidean space, but T is imaginary. So, if we assume that coordinates are complex, the only difference between euclidaen space and Minkowsky space is an orientation of a subset in a 4-dimensional complex space.

So far I hope it is correct.

My question is, what about General relativity and curved spaces? I read that for 3space+1time curved space can be put into 86 dimensional manifold with 3 timelike dimensions.

What if we work completely in the complex area, so there is no difference between space and time dimensions?