In writing a real matrix $A$ with non-real eigenvalues in the form $A=PCP^{-1}$, we find two eigenvalues: $\lambda$ and $\overline{\lambda}$. We can find a vector $\mathbf{v}\in\mathbb{C}^2$ corresponding to the eigenvalue $\lambda$, and then the vector $P$ can be $\begin{bmatrix}Re(\mathbf{v}) & Im(\mathbf{v})\end{bmatrix}$. At that point, our matrix $C$ is $\begin{bmatrix} Re(\overline{\lambda}) & -Im(\overline{\lambda}) \\ Im(\overline{\lambda}) & Re(\overline{\lambda})\end{bmatrix}$. Great.
Alternatively, if we find a vector $\mathbf{v}\in\mathbb{C}^2$ corresponding to the eigenvalue $\overline{\lambda}$, then we use it to write $P$ as above, but we use $C = \begin{bmatrix} Re(\lambda) & -Im(\lambda) \\ Im(\lambda) & Re(\lambda)\end{bmatrix}$.
I'm looking for the most intuitive way to explain why we use one of the two eigenvalues to write down matrix $C$, but the other one - its complex conjugate - to find an eigenvector for writing $P$.
Any insight, especially explanations that will resonate with first-time linear algebra students, would be very welcome.