(I also think that without the geometric perspective complex numbers hardly make any sense at all.)
Here's a fairly typical complex numbers problem (from a PARCC Algebra II sample test):
A fundamentally different approach involves understanding how multiplication by a complex numbers encodes rotations and scalings.
The vast majority of mistakes that students currently make with complex numbers are algebraic. This speaks volumes about how our kids are generally taught and the limitations of doing so. To really understand complex numbers with any depth is to understand them both algebraically and geometrically.