I vote for “complete” numbers in place of “complex”, and we just don’t reference “imaginary” at all — it goes from “real” to “complete” when you add the missing root of unity.
A complex number should be thought of as the quotient of plane vectors: a quantity z = v/u which when multiplied by vector u yields v: that is zu = v.
(Important note: multiplication of a complex number by a vector is not commutative, zu ≠ uz.)
For more: http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
I would philosophically say "Completeness" is a more important property (since now negative roots have a meaning attached to them). But their rotational aspect is perhaps more mind blowing.
edit: to match with existing terminology you might call the reals the "completes" and the complex numbers the "algebraically-closeds" but... meh.