i'm learning about this right here as I read, but do you mean complex numbers rather than imaginary?

Same question.

Or more generally, vectors. They don't have a total order, because as we define "less than"/"greater than" in terms of magnitude (length), this means for any vector V (other than 0), there's an infinitely many vectors that are not equal to V, but whose length is equal to length of V.

Is this is what ≹ is talking about?

Yep, that’s what I meant, sorry!

As I've written in another comment here, a great example of a number-y field which is not totally ordered is Games ⊂ Surreal Numbers ⊂ ℝ. There you have certain "numbers" which can can confused with (read incomparable) whole intervals of numbers. Games are really cool :)

You've got the subset relationships backwards: Reals are a subset of the field of Surreal Numbers which is a subset of the group of Games. (Probably better phrased as embeddings, rather than subsets, but the point remains...)

Note that Games do _not_ form a field: there is no general multiplication operation between arbitrary games.