Let $\mathfrak h_t$ be the KPZ fixed point started from any initial condition that guarantees $\mathfrak h_t$ has a maximum at every time $t$ almost surely. For any fixed $t$, almost surely $\max \mathfrak h_t$ is uniquely attained. However, there are exceptional times $t \in (0, \infty)$ when $\max \mathfrak h_t$ is achieved at multiple points. Let $\mathcal T_k \subset (0, \infty)$ denote the set of times when $\max \mathfrak h_t$ is achieved at exactly $k$ points. We show that almost surely $\mathcal T_2$ has Hausdorff dimension $2/3$ and is dense, $\mathcal T_3$ has Hausdorff dimension $1/3$ and is dense, $\mathcal T_4$ has Hausdorff dimension $0$, and there are no times when $\max \mathfrak h_t$ is achieved at $5$ or more points. This resolves two conjectures of Corwin, Hammond, Hegde, and Matetski.