For a linear autonomic differential equation of the plane $$\dot x = Ax,$$ with $A ∈ \operatorname{Mat}_{2×2} (ℝ)$, say we have a fundamental matrix $Φ \colon ℝ → \operatorname{Mat}_{2×2} (ℝ)$, that is an invertible solution to the differential equation $$\dot X = A X$$ as an equation on $\operatorname{Mat}_{2×2} (ℝ)$.
If $Φ$ is conjugate over $ℂ$ to some $Ψ \colon ℝ → \operatorname{Mat}_{2×2} (ℂ)$ with $$Ψ = \begin{bmatrix}\exp λ t & 0 \\ 0 & \exp \overline λ t \end{bmatrix}$$ for some $λ ∈ ℂ$ (and $λ \notin ℝ$), how can we deduce something about the quality of $Φ$?
I mean, if I could think visually in four dimension, it would be pretty clear how the solutions $$\begin{bmatrix} \exp λ t \\ 0\end{bmatrix}\quad\text{and}\quad\begin{bmatrix} 0 \\ \exp \overline λ t\end{bmatrix}$$ would look like. And I know that $\exp λt$ and $\exp \overline λ t$ in $ℂ$ “pretty much look like” rotations.
But how can I deduce from this that the solutions in $Φ$ “pretty much look like” rotations as well?