Here is my question:
We all now that a line $d$ can be written as:
$d:\begin{pmatrix}x\\y\\ \end{pmatrix} = \begin{pmatrix}a_1\\a_2\end{pmatrix} + \lambda\begin{pmatrix}v_1\\v_2\\ \end{pmatrix}, \lambda\in\mathbb{R}$
where $a_1, a_2$ are the coordinate of a point $A \in R^2$ in this case and $v_1, v_2$ are the coordinate of the direction vector.
My question is what the parametric equation $d$ represent if $\lambda \in\mathbb{C}$
If you write $\lambda=re^{i\theta}$, i.e. in polar form, and make the association $$ (a_1,a_2)\mapsto a_1+ia_2\\ (v_1,v_2)\mapsto v_1+iv_2\\ $$ then you have that $x+iy$ is $a_1+ia_2$ plus the vector that is $v_1+iv_2$ rotated through the angle $\theta$ and scaled by $r$.