[halayli]

My naive take is we discovered it as a math tool first but later on rediscovered it in nature when we discovered the electromagnetic field.

The electromagnetic field is naturally a single complex valued object(Riemann/Silberstein F = E + i cB), and of course Maxwell's equations collapse into a single equation for this complex field. The symmetry group of electromagnetism and more specifically, the duality rotation between E and B is U(1), which is also the unit circle in the complex plane.