Usually, those complex terms come from treating time as imaginary. THis is frequently done in SR, but is rarely done in GR, where one usually sees the covariant derivative.
There are occasions when one does use complex coordinates in GR - I've never done it. (It's common enough that GRtensor provides allowanes for it, though).
As far as I know it doesn't affect taking the covariant derivative at all there are several ways of looking at it, the way I prefer to describe the covariant derivative is that it's just like the ordinary derivative, except that you have to parallel transport vectors (and/or tensors) to the same location before you subtract them and divide by the delta.