In his first proof of the Fundamental Theorem of Algebra, Gauss deliberately avoided using imaginaries . When formulated for a polynomial with real coefficients, the theorem states that every such polynomial can be represented as a product of first and second degree terms. Second degree factors correspond to pairs of conjugate complex roots. The latter formulation is not only more elegant, it's also more revealing. Staying in the realm of real numbers it's hard to explain why and wherefrom quadratic terms appear.
Complex numbers provide an immediate explanation.
Complex numbers indeed proved to be a natural setting for the theorem. But the realization of course did not come immediately. Real functions may or may not have derivatives. Furthermore, existence of a derivative in one point does not assure its existence anywhere else. A real function may have one or two or, for that matter, any finite number of derivatives.