That's not true: the multiplication is different. (x1,y1) × (x2,y2) would have to be (x1x2 - y1y2, x1y2 + y1x2), not (x1y1, x2y2). The addition does work as

That's not true: the multiplication is different. (x1,y1) × (x2,y2) would have to be (x1x2 - y1y2, x1y2 + y1x2), not (x1y1, x2y2). The addition does work as you suggest, so a more correct formal statement would be: the complex numbers form an additive group that is isomorphic to the cartesian product of the additive group of real numbers and itself; the complex numbers also happen to form a ring (indeed a field).