The fields of the surreal and hyperreal numbers aren’t complete. I’ve noticed that $\mathbb{Q}^2$ isn’t rotationally complete as $(0,1)$ could be rotated to a point not in the rational plane (like $(1,1)/\sqrt{2}$). This is because the field itself isn’t metrically complete and I’m wondering if this causes $No^2$ or $(\mathbb{R}^*)^2$ to have the same issues or if the incompleteness is such that it isn’t (I’m thinking of how two intervals of the real line form an annulus as the surreal and hyperreal lines are made of many connected components).
So given the two planes, are they invariant under rotations like the standard Euclidean plane is?