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.