I've encountered a notion of distance (their usage) in the field of ecology that raises an eyebrow, and would like to know if it's used elsewhere (if it makes any sense at all).
If you diagonalize a real symmetric $nxn$ matrix and then scale each eigenvector by the square root of its associated eigenvalue (see PCoA), you may have earned yourself an orthogonal basis for a complex space. Can you get away with defining a (non-metric) distance, $d_{ij}$, between points $z_i = \{a_{ik} + b_{ik}i\}_{k\in n}$ that satisfies:
$$ d_{ij}^2 = \sum_{k=1}^n (a_{ik} - a_{jk})^2 - (b_{ik}-b_{jk})^2 $$
Yes, minus. If you can, and it's a thing, what's it called?
Edit: Thanks, Servaes. Fixed.