In complex numbers the module can be defined as
$$|z|=\exp(\Re(\ln z))$$
I wonder, to what extent a similar value is useful/used/meaningful in other rings? I mean, instead of taking the real part $\Re$ one may use trace or other means of obtaining the "real part" of an object. Does it still hold meaningful?
What if it turns out that $|z|<\Re (z)$ in some field? Obviously, the Pythagorean theorem gets broken and such "module" cannot be a norm, but otherwise, what properties does it still keep?