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.