The transformations of a quaternion are best described by matrix multiplication, as are the transformations of any other hyper-complex number system I’ve heard of.
Is there an analog of the quaternion group that can only properly be described using tensors?
The group law on the Brauer group ${\rm Br}(K)$ of a field $K$ is based on the tensor product: for two finite-dimensional central simple $K$-algerbas $A$ and $B$, the product of the equivalence classes $[A]$ and $[B]$ in ${\rm Br}(K)$ is given by $$ [A][B] = [A \otimes_K B]. $$ The inverse of $[A]$ is $[A^{\rm op}]$, where $A^{\rm op}$ is the opposite algebra to $A$ and this operation on ${\rm Br}(K)$ is associative because of the associativity of tensor products of algebras (up to isomorphism).
What does this have to the with the Hamilton quaternions $\mathbf H$? That's essentially the nontrivial element of ${\rm Br}(\mathbf R)$, and more generally a quaternion algebra over a field $K$ is essentially an element of order $2$ in ${\rm Br}(K)$.