Boris Zilber has argued that the field of the complex numbers is "logically perfect". For one thing, the theory of an algebraically closed field of characteristic zero is uncountably categorical: it admits a unique model, up to isomorphism, of some uncountable cardinality - and thus a unique model of each uncountable cardinality, by Morley's theorem!
The field of real numbers is not "logically perfect" in Zilber's sense. More precisely, the theory of a real closed field is not uncountably categorical.
This led Terry Bollinger to ask about the quaternions.
Has anyone studied this? For starters, we need to settle on a 'theory of quaternions'. I want a theory where the only operations are + and ×, and the only constants are 0 and 1. Here's my guess about how to proceed. First we impose the axioms of a division ring (including associativity for multiplication, to rule out the octonions!). We impose a 'characteristic zero' axiom schema that says $1 + 1 + \cdots + 1$ can never equal zero. Then we impose an 'algebraically closure' axiom saying that any polynomial whose coefficients all commute has a root. (Any commuting set of quaternions lies in a subring isomorphic to the complex numbers.) Finally, we impose an axiom saying that multiplication is not commtuative, to rule out the complex numbers.
I haven't thought about this much, but I'm hoping these are enough, with the help of the Frobenius theorem.
Then we can ask if the theory of quaternions is uncountably categorical. My guess is that it's not.
Here's my rough reasoning. The center of the quaternions forms a copy of the real numbers. Moreoever, I'm guessing that we can prove using our 'theory of quaternions" that the center of the quaternions is a real closed field. Conversely, starting from any real closed field, we can build a model of the theory of quaternions. So, I believe we're in trouble, since the theory of a real closed field is not uncountably categorical.
Perhaps someone can say if this sort of argument is valid: if in some structure we can define a subset and this subset, equipped with some operations in our theory, is a model of a second theory that's not uncountably categorical... and any model of that second theory arises in this way... then the original theory could not have been uncountably categorical.
Or, if this doesn't work, maybe someone can straighten things out!