In addition to the two reasonably well-known categories $\mathrm{SuperVect}_{\mathbb R}$ and $\mathrm{SuperVect}_{\mathbb C}$ of *real* and *complex* *super vector spaces*, each of which is monoidally equivalent to corresponding category of $\mathbb Z/2$-graded vector spaces but with the Koszul sign rules, there is a third much less well-known symmetric monoidal category that I like to call the *quaternionic super vector spaces* $\mathrm{SuperVect}_{\mathbb H}$. It appears, among other places, when studying the statistics of a certain type of pinor.

As a category, $$ \mathrm{SuperVect}_{\mathbb H} = \mathrm{Vect}_{\mathbb R} \oplus \mathrm{Mod}_{\mathbb H} $$ The monoidal structure is a bit funny, using the Morita equivalence $\mathbb H \otimes \mathbb H \simeq \mathbb R$. Here is a description of it. Recall that the usual Galois correspondence between $\mathbb R$ and $\mathbb C$ identifies $\mathrm{Vect}_{\mathbb R}$ with the category of complex vector spaces $V_{\mathbb C}$ equipped with an antilinear involution, i.e. $\varphi: V_{\mathbb C} \to V_{\mathbb C}^*$, $\varphi^*\varphi = 1$. Similarly, one can identify $\mathrm{Mod}_{\mathbb H}$ with the category of complex vector spaces $V_{\mathbb C}$ equipped with an antilinear "antiinvolution", i.e. $\varphi: V_{\mathbb C} \to V_{\mathbb C}^*$, $\varphi^*\varphi = -1$. (By definition, $V^* = V$ as real vector spaces, but the $\mathbb C$-action is that $\lambda \in \mathbb C$ acting on $v\in V^*$ is given by the action $v\lambda^*$ in $V$. So $\varphi^* = \varphi$ as real linear maps, but I'm thinking of them as $\mathbb C$-linear in two different ways.) Using this, you can identify $\mathrm{SuperVect}_{\mathbb H}$ with the category of complex supervector spaces equipped with an antilinear map that squares to $1$ on the even part and to $-1$ on the odd part. If you check carefully, you'll see that the tensor product (in $\mathrm{SuperVect}_{\mathbb C}$) of two such objects is naturally such an object, and in this way you can recover the symmetric monoidal structure on $\mathrm{SuperVect}_{\mathbb H}$.

In any sufficiently nice linear category (and $\mathrm{SuperVect}_{\mathbb H}$ is plenty nice for these purposes) you can develop a theory of associative algebras, bimodules, and Morita equivalence. Among other things, you can define a *Brauer group* for your given category whose elements are Morita equivalence classes of Morita-invertible algebras. (I.e. the group of units of the monoid whose objects are Morita equivalence classes of algebras and whose multiplication is tensor product.)

It is well known that the Brauer groups in this sense of $\mathrm{SuperVect}_{\mathbb C}$ and $\mathrm{SuperVect}_{\mathbb R}$ are respectively $\mathbb Z/2$ and $\mathbb Z/8$, and that this is closely related to periodicity in various K-theories.

Question:What is the Brauer group of $\mathrm{SuperVect}_{\mathbb H}$? What are the simple representatives of the elements? What "K-theory" is it related to?