A categorification of the square root of -1

We give a graphical calculus for a monoidal DG category $\mathcal{I}$ whose Grothendieck group is isomorphic to the ring $\mathbb{Z}[\sqrt{-1}]$. We construct a categorical action of $\mathcal{I}$ which lifts the action of $\mathbb{Z}[\sqrt{-1}]$ on ${\mathbb{Z}}^2$.