One nice way of seeing the inevitability of the complex numbers is to view them as a metric completion of an algebraic closure rather than a closure of a completion.

Taking the algebraic closure of Q gives us algebraic numbers, which are a very natural object to consider. If we lived in an alternative timeline where analysis was never invented and we only thought about polynomials with rational coefficients, you’d still end up inventing them.

If you then take the metric completion of algebraic numbers, you get the complex numbers.

This is sort of a surprising fact if you think about it! the usual construction of complex numbers adds in a bunch of limit points and then solutions to polynomial equations involving those limit points, which at first glance seems like it could give a different result then adding those limit points after solutions.