[jonahx]

But you can define the complex field C. And it has many benefits, like making the fundamental theorem of algebra work out. I'm not seeing the issue?

On a similar note, why insist that "i" (or a negative, for that matter) is an "attribute" on a number rather than an extension of the concept of number? In one sense, this is a just a definitional choice, so I don't think either conception is right or wrong. But I'm still not getting your preference for the attribute perspective. If anything, especially in the case of negative numbers, it seems less elegant than just allowing the negatives to be numbers?

[ActorNightly]

Sure, you can define any field to make your math work out. None of the interpretations are wrong per say, the question is whether or not they are useful.

The point of contention that leads to 3 interpretations is whether you assume i acts like a number. My argument is that people generally answer yes, because of Eulers identity (which is often stated as example of mathematical beauty).

My argument is that i does not act like a number, it acts more like an operator. And with i being an operator, C is not really a thing.