I was in my Complex Analysis class, and the professor said that we should look for a field, rather than a group, to do calculus over. Why is this the case? I understand that we gain another operation with a field, but if it's just a matter of gaining another operation, why not use some object with three operations. (I've never learned of one.)
So why do we seem to want fields, like $\mathbb{R}$ and $\mathbb{C}$ for performing calculus over? Is there something about them that makes analysis work, or are they simply a richer subject of study? If the latter, why not include more operations?