Quaternions—in the sense of objects that obey the relations of the quaternion group—arise automatically in Lie Theory. The most elementary noncommutative continuous group is $SO(3)$, and if you want to understand rotations in three dimension, you obviously have to understand the representations of this group—how it can act on scalar, vector, and higher tensor objects, for instance. The most natural way to do this is to study the Lie algebra; even if you are not going to fully generalize to abstract Lie algebras, you are going to be led to a study of the generators $T_{i}$, which obey a commutation relation $[T_{i},T_{j}]=\epsilon_{ijk}T_{k}$.
However, if you explore the representations of this Lie algebra, $\mathfrak{so}(3)$, you find that there are a whole bunch of group representations missing—just like there are missing roots to real polynomials! There is one finite-dimensional irreducible representation of the algebra $\mathfrak{so}(3)$ for dimension $n$ for each integer $n\geq1$. Yet the group $SO(3)$ only has representations with odd dimension! We know the reason for this now is that there is a topological obstruction; $SO(3)$ is not simply connected. However, even without understanding the general topological issue involved, it straightforward to locate another group—the universal cover $SU(2)$ of $SO(3)$—which has the same Lie algebra $\mathfrak{su}(2)\approx\mathfrak{so}(3)$ but which does have all the representations, including a two-dimensional fundamental representation. The generators of this two-dimensional representation (from which, via tensor products, all the representations can be built up) $$i\sigma_{1}=\left[\begin{array}{cc} 0 & i \\ i & 0 \end{array}\right] \\ i\sigma_{2}=\left[\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right] \\ i\sigma_{1}=\left[\begin{array}{cc} i & 0 \\ 0 & -i \end{array}\right]$$ then have precisely the multiplication table of the quaternions $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$.
Of course, the connection between the quaternion algebra and the theory of continuous groups goes on to be much deeper. However, this is the first place that it is likely to be evident that the introduction of objects that obey the defining relations for quaternions is necessary.