Is there a geometry where the distance between two points can be complex? This tweet contained this image which is of course complete nonsense but it got me thinking -- is there such a thing where the distance of two points is a complex number?

Ps. it seems such questions are fit for this SE for example Can there be a geometry where angles can be infinite?
 A: Akbar Azam, Brian Fisher & M. Khan (2011) Common Fixed Point Theorems in Complex Valued Metric Spaces, Numerical Functional Analysis and Optimization, 32:3, 243-253, DOI: 10.1080/01630563.2011.533046

We introduce complex valued metric spaces and obtain sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type conditions.

While I can't find this paper freely available online, this PDF cites it and defines " XxX→C is called a complex valued b-metric":

A: There can be no such metric, as it would not obey the requirements of a metric.  Specifically, you must be able to have the "$>$" operation (for the triangle inequality: $d(a,b) + d(b,c) \geq d(a,c)$), and that does not hold for complex numbers.
A: A distance is always a real number, for reasons given in other answers. It is only when the angle is also introduced to make it a vector that things get complex. For example the shortest distance from one point to another is a real number, but the shortest path is a vector and may prove to be complex.
