Are there different products that turn $\,\Bbb R^2$ into a field?

Question

With this product:

$$(a,b)\cdot (x,y):=(ax-by,ay+bx)$$

$\,\Bbb R^2$ becomes a field, actually isomorphic to $\,\Bbb C$.

Are there other interesting products that turn $\,\Bbb R^2$ into a field?