  Let $L$ be the language of rings. We provide an axiomatization of the
$L$-theories of quaternions and octonions and we characterize the models of
mentioned theories: they coincide, up to isomorphism, to quaternion and
octonion algebras over a real closed field, respectively. We prove these
theories are complete, model complete and they do not have quantifier
elimination. Then, we focus on the class of ordered polynomials. Over
$\mathbb{H}$ and $\mathbb{O}$ these polynomials are of special interest in
hypercomplex analysis since they are slice regular. We deduce some fundamental
properties of the zero loci of ordered polynomials from completeness and we
prove the failure of quantifier elimination for the fragment of ordered
formulas.
