Historically the square root of negative one arose more naturally than you might think; it is basically impossible to avoid it when you derive a cubic formula.

Historically the square root of negative one arose more naturally than you might think; it is basically impossible to avoid it when you derive a cubic formula. Historically the work on complex numbers started as soon as cubic and quartic formulas were derived, because complex numbers emerge naturally and unavoidably from those formulas (even for polynomials that have only real roots). At first mathematicians were generally dismissive of the imaginary units and viewed them as "nonsense" terms that sometimes appear in some formulas and there was a lot of doubt about dealing with such terms (the same doubt you might have if someone divided by zero while solving some formula, even if they wind up finding the correct answer). Those doubts were completely eliminated by the proof of the fundamental theorem of algebra. For what it's worth, you can avoid "imaginary" numbers entirely if you want -- you can do arithmetic on real polynomials modulo x^2+1, or equivalently on 2x2 real matrices of a special form.