What is this 2D division algebra?
Solution 1:
The two-dimensional division algebras over the real numbers were completely classified by Burdujan in 1985. See for example section 2 of this article for a short review. Borrowing notation from the article, we have that every such algebra is isomorphic to $\mathbb{C}_{ST}$, the complex numbers equipped with a modified multiplication $x \circ y = S(x)\cdot T(y)$, where $S, T$ are invertible $\mathbb{R}$-linear transformations of $\mathbb{C}$.
For the division algebra in your question, $S$ is complex conjugation and $T$ the identity.