This is an easier version of a more general question I proposed, which hasn't received much attention. How many binary operations can we assign to $\mathbb R$ which make it into a group, where group multiplication and inversion are continuous with respect to the euclidean topology? I'll get us started:
We can define $x*y = x + y$ (standard addition). But we could also use $x*y = h^{-1}(h(x)*h(y))$ for any homeomorphism from $\mathbb R$ into $\mathbb R$. And this satisfies the group axioms because of bijectivity and is continuous since $h$ and $h^{-1}$ are.
There might be a simpler answer, but by Hilbert's 5th problem any topological manifold group is a Lie group, and there is but one 1dimensional 1-connected Lie group, namely $(\mathbb R,+)$. So for any continuous group operation $*$, the topological group $(\mathbb R,*)$ is isomorphic to $(\mathbb R+)$, i.e. there is a homeomorphism $\phi:\mathbb R\to\mathbb R$ s.t. $x*y=\phi^{-1}(\phi(x)+\phi(y))$.