How does Ohm's law extend to Z if Z is complex?
Complex doesn't mean unearthly.

When you say Z is complex. That means there will be a phase shift between the V and I components.
Complex notations helps represent that phase relationship between the V and I (simply called phase shift) in a understandable/manageable way.

For a purely ohmic impedance( represented with a real number). The V and I remain in-phase.