In line with Joel’s answer, we have all sorts of uniqueness of ultrapower results that follow form CH and whose negations are known to be consistent with ZFC. Since classical model theory is not my area, I’ll let others add those examples. For me, I’ll point out an example in my field: For any separable (or, more generally, of density character less than or equal to continuum, under the $\sigma$-strong topology) II$_1$ factor $M$ (i.e., a tracial von Neumann algebra with scalars as its center), its ultrapower $M^\mathcal{U}$ is independent of the choice of free ultrafilter $\mathcal{U}$ on $\mathbb{N}$ iff CH holds.

In line with Joel’s answer, we have all sorts of uniqueness of ultrapower results that follow form CH and whose negations are known to be consistent with ZFC. Since classical model theory is not my area, I’ll let others add those examples. For me, I’ll point out an example in my field: For any separable (or, more generally, of density character less than or equal to continuum, under the $\sigma$-strong topology) II$_1$ factor $M$ (i.e., an infinite-dimensional tracial von Neumann algebra with scalars as its center), its ultrapower $M^\mathcal{U}$ is independent of the choice of free ultrafilter $\mathcal{U}$ on $\mathbb{N}$ iff CH holds.