The Complex numbers form what is called a Field in the theory of groups.
Satisfying the 11 properties of a Field, albeit not being an ordered field in that the square power of the imaginary unit is -1. They do not hold the trichotomy property.
This will result that imaginaries are not considered greater than or less than other elements of the universe set of pure imaginary or complex numbers. They are considered only as Vectors with Magnitudes.
Because Closure, Assossiativity and Commutivity under two Operations; Addition and Multiplication namely, and Existence of Identity Elements for Addition and Multiplication along with inverse elements for these respective Operations, however excluding 0 + 0i, and additionally that Complex Numbers are Distributive with respect to Multiplication over Addition, we can define in algebraic terms that every operation on dividing by a complex number will yield a complex number belonging to the Universal Set of Complex Numbers.
This is why the method works in the sense you will always produce a real solution or complex solution to a division problem with complex numbers in the denominator. The algebra forces a functional output belonging to the Complex Numbers in every infinitely possible case.
It is a complicated way of saying, "When adding apples and oranges, (or multiplying them), you are always left with some countable discreet number of apples and oranges, no bananas allowed!" (This is why they do not include division by zero.)