Properties lost when going from real number system to quaternions and octonions

Question

What properties do we lose as we go from real numbers to quaternions, then to octonions? Do any new properties arise, or do calculations just become more "path dependant"?

Answer 1

From reals to complex you lose order. From complex to quaternions you lose commutativity. From quaternions to octonions you lose associativity. From octonions to ..?

I had written this as a comment, then I followed Noah Schweber's link, which essentially says this plus more. And pregunton's answer is meaty.
What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

EDIT: Bcpicao's comment led me to
https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction
a fabulous link.

Answer 2

The most important property (IMHO) that $\mathbb{R}$ has that its extensions don't have is that it is a totally ordered field with least upper bound property. You don't have an order (that works nicely with the field) in these extensions.