Recently, Joel David Hamkins presented a historical thought experiment that shows that CH could have been adopted as an axiom if we had been using the hyperreal field $\mathbb{R}^*$ instead of $\mathbb{R}$ in calculus, since we'd like to have a categoricity result for $\mathbb{R}^*$ in this alternative history.

Regardless of the validity of this argument, which I am not planning to discuss here, I thought it would be nice to have a collection of uniqueness results that follow from CH, not to claim that CH should be an axiom but simply to have such a list for its own sake. There seems to be no similar question on MO as far as I can tell. Therefore, my big-list question is:

What are some uniqueness results (whose negations are believed/known to be relatively consistent with ZFC) that follow from CH?