Representing a complex line as a directed ellipse

Consider nonzero $v = v_r + iv_i \in \mathbb{C}^n$, It can be thought of as an ordered 2-tuple of vectors $(v_r, v_i)\in \mathbb{R}^n\times\mathbb{R}^n$.

The complex line generated by $v$ is
$$\{r[(\cos(\theta) v_r -\sin(\theta) v_i) + i(\sin(\theta) v_r +\cos(\theta) v_i)]:\\ r\ge 0, \theta\in[0, 2\pi]\}$$
So, essentially, it is a set of concentric ellipses. We consider one of them:
$$\{(\cos(\theta) v_r -\sin(\theta) v_i) + i(\sin(\theta) v_r +\cos(\theta) v_i): \theta\in[0, 2\pi]\}$$
As $\theta$ increases, both $(\cos(\theta) v_r -\sin(\theta) v_i)$ and $(\sin(\theta) v_r +\cos(\theta) v_i)$ rotate in the ellipse, always being conjugate to each other.

So we could represent the complex line generated by $v$ as a directed ellipse, with the direction determined by $v_r$ "pointing counterclockwise towards" $v_i$.

Multiplying $v$ by $e^{i\theta}$, then, is to rotate the conjugate pair on the ellipse clockwise by $\theta$.

Conjugating the complex line is then reversing the direction of the ellipse. Conjugation does not change the line iff the ellipse is degenerate, that is, $v_r, v_i$ are $\mathbb{R}$-linearly dependent.

I found this perspective very new and helpful, but I have never seen this presented this way before. I asked on math.stackexchange but nobody replied. Would anyone point me to a reference?