In Physics the complex i is used for different purposes, which makes it seem more complex that it really is. My favorite explanation of complex numbers is the paper "Imaginary Numbers are not Real – the Geometric Algebra of Spacetime" [1] which shows what the complex i actually is in terms of Clifford Algebras.

[1] http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are...