I recently asked a queston regarding the proper, nontrivial ideals of what I called the modular Lipschitz quaternions, which was part of my series of open problems for enthusiasts. As luck would have it, I realized only after the fact that my key assertion--the quaternion ring $Q_p$, where $p$ is prime, is a field if it is a domain--is wrong. The specific issue is the inapplicability of Wedderburn's theorem, since it is easy to show every quaternion ring of characteristic $>2$, even over a general coefficient ring $R$, is noncommutative (thus cannot possibly be a field!). Thus the work done by @rschwieb and @MatheiBoulomenos shows that none of the $Q_n$ are domains (which, as @MarianoSuárez-Álvarez pointed out, is actually a simple consequence of Wedderburn's little theorem in each case but $Q_2$) but, due entirely to my error, failed to answer the original question. In order not to edit all meaning out of the previous post, I have decide to create this post as a new home for the questions. To wit, these are
For which $n$, if any, does $Q_n$ have at least one proper nontrivial ideal?
If $Q_n$ has proper nontrivial ideals what are their orders? How, if at all, are they related to the ideals of $\mathbb{Z}_n$?
As remarked by @rschwieb, it is obvious that an ideal of $\mathbb{Z}_n$ generates an ideal of $Q_n$. The converse does not hold (at least for general $n$), as @JyrkiLahtonen has identified a proper, nontrivial ideal of $Q_2$.
Although an initial attempt at a proof by myself was not successful (as pointed out by @JyrkiLahtonen, whose comments are of interest even taken on their own), it yielded two lemmas which may or may not be helpful. We use the following notation: if $A$ is a set of quaternions $P_1A$ is the set of coordinates of 1 contained in $A$, $P_iA$ the set of coordinates for $i$, and so on. We will call these the unit projections of $A$.
Lemma 1. If $S$ is an additive subgroup of $Q_n$ then the unit projections of $S$ are (possibly different) additive subgroups of $\mathbb{Z}_n.$
Computing the unit projections of $q-p$, for $q,p\in S$ shows this immediately.
Lemma 2. If $I$ is an ideal of $Q_n$, then the unit projections of $I$ are the same subgroup of $\mathbb{Z}_n$.
Proof: If $I$ is an ideal it is, in particular, an additive subgroup of $Q_n$. Lemma 1 implies the unit projections of $I$ are additive subgroups of $\mathbb{Z}_n$. To show these groups are the same, we need only show they contain the same members. For instance, suppose $a \in P_1I$. It follows that there is a quaternion $q \in I$ for which $P_1q = a$. Since $I$ is an ideal, it is closed under multiplication by the elements of $Q[n]$, in particular $i$, thus $iq \in I$ and therefore $a \in P_iI$. On the other hand, if $a \in P_iI$ then multiplication by $-i$ shows $a \in P_1I$. The other equalities are established in a similar way.
New Questions
Finally, I leave the community with two new questions related to this topic.
Classify the ideals of $Q[R]$, where $R$ is an arbitrary ring. Particularly, we may suppose $R$ is finite.
This Wikipedia article partially documents work on the Hurwitz and Lipschitz quaternions. One question I find particularly compelling is whether or not a similar theory of irreducible factorization can be created for $Q_n$ or $Q[R]$.