Let $K$ be a finite field extension of $\mathbb{Q}$. Let $H_K:=K \oplus Ki \oplus Kj \oplus Kk$, where $i,j,k$ satisfy $i^2=j^2=k^2=ijk=-1$ as for the usual quaternions but this time over the field $K$.
Question 1: What are the elements $x \in H_K$ that satisfy a polynomial equation $f(x)=0$ where $f(x)$ is monic and has integer coefficients? Call such elements $x$ integral.
Let $R_K$ be the ring generated by such integral elements $x$.
Question 2: What are the units of $R_K$?
Special cases are of course also interesting, like quadratic extensions of $\mathbb{Q}$.