The topological space $\mathbb{P}^1_{\mathbb{R}}$ is a circle, so it has an abelian group structure just defined by adding angles but this is not algebraic, not compatible with the multiplication you describe, and it can't be extended to $\mathbb{P}^1_{\mathbb{C}}$ (which is a sphere and thus has no topological group structure), nor does it restrict to $\mathbb{P}^1_{\mathbb{Q}}$. EDIT: As Noam Elkies points out, the circle group structure can be made algebraic on $\mathbb{P}^1_{\mathbb{R}}$ by defining it by $\tan(x+y)=\tan(x)+\tan(y)$, instead of by adding angles. This still has all the other defects listed above.
More generally, it's impossible to do this algebraically: it's an extremely well known fact that a projective curve with an algebraic group structure (as addition would be) must be an elliptic curve.