In flat Minkowski space with the metric signature (-,+,+,+) the spacetime interval $ds^2$ is defined as
$ds^2 = -c^2 dt^2 + x^2 + y^2 + z^2$
This interval (or better the square of the distance) can become negative (for timelike intervals) so the spacetime distance $ds$ can be complex.
In quantum mechanics a complex quantity is something which cannot be measured. Only real quantities can be measured. Does that mean that the spacetime distance becomes unmeasurable if it becomes complex? Or how is the interpretation in relativity for complex spacetime distances?
What confuses me is that one can also choose a different metric $(+,-,-,-)$. In that case the spacetime interval is
$ds^2 = c^2 dt^2 - x^2 - y^2 - z^2$
Then it is not timelike distances that become complex, but instead spacelike distances will be complex. So it seems to depend ob the convention of the metric signature when the space-time distance will be complex. So do complex spacetime distances (or negative spacetime intervals) have any physical meaning at all?