This is total shit quickly banged out in probably hard to follow semi-formalism.
Complex numbers (C) can be viewed as a scalar bivector pair. A 2D bivector requires 1 element (note 1), so we have C ~= R1+R1 ~= R2 with basis {1,i}.
Denote a complex number: z = a+bi = (a, b), {a,b} on R. Associating C with R2 we can interpret (a,b) as a coordinate (in the plane). Functions over C map coordinates in the plane to other coordinates in the plane. We can define principle value (single valued) functions for: log, exp and powers (among others).
For power we can rewrite 'z' into a polar form z = m(cos(a), sin(a)), m >= 0, a on [pi,-pi). Then the princle power can be denoted as:
zt = mt(cos(ta), sin(ta))