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I expect that it is because in general, more useful functions and operations are closed in the complex plane. Things that may superficially appear complicated to a human being learning math, like analytic extension [1], from a theoretical perspective means that when you need the tool, it is there, and continues to expand the sets of problems you can address. As you add restrictions to the set of numbers you want to use, you are importing those restrictions into every proof you want to do on those numbers. For instance, consider the question, "Does this polynomial of some high degree have roots, and if so, how many?" The Fundamental Theorem of Algebra proves that the answer is yes, and the number of roots is the same as the degree (although some may be roots multiple times). See [2]. Now, as you watch 2, consider what happens to the proof if you confine it even to the reals, let alone the integers. While the complex plane is "more complex" from a human perspective, the proof of the Fundamental Theorem of Algebra using the complex plane is not that complicated. Consider trying to make equivalent statements about polynomials with only real roots, and not using the complex plane in the proofs. In cases where it is feasible, your proofs will inevitably be carrying around a lot more caveats about what polynomials it applies to, and there may be things that the caveats simply render infeasible. As you step down the number hierarchy, the caveats get worse and worse. There's more and
