[zero-sharp]

The complex numbers are vectors with a special kind of product which turns them into a field (an algebraic structure that behaves like a "number"). If you think about it that way, then it doesn't really seem so mysterious. There are all sorts of weird algebraic structures that have similar properties to "numbers". Not all structures have all the properties you want, nor do they all extend the real numbers though.

I don't know much about physics, so I actually have no idea if imaginary quantities are observable, or if they're just a purely formal mathematical device. Or I guess another question is: are there physical relationships we cannot state without the complex numbers? If you can rewrite a statement about a complex number in terms of a vector, you can imagine that a lot of work can just be rewritten to deal entirely with real number pairs instead? That's how I would judge if something is "real".

https://physics.stackexchange.com/questions/11396/can-one-do...

https://physics.stackexchange.com/questions/32422/qm-without...

[FillMaths]

That's similar to what the author says in the second paragraph. But he goes on to consider many other subtle notions arising from the fact that the complex field is not rigid. How can we tell i from -i? They have all the same properties with respect to the field structure.