The complex slope $\mu$ of a line containing the points $z_1$ and $z_2$ in the complex plane is defined as $\frac{z_1-z_2}{{\displaystyle \stackrel{-}{z_1}}-z_2}.\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}If\hspace{0.17em}\hspace{0.17em} }{\mu }_1,{\mu }_2$ are the complex slopes of two lines $L_1$ and $L_2,$ then

We observe that if $z_0$ is a non – zero complex number and c is a real number, then the equation ${\overline{z}}_{0}z+z_0\overline{z}+c=0$ represents a straight line with complex slope $\frac{-z_0}{{\overline{z}}_0}$  
Let $L_1\overline{\alpha }z+\alpha \overline{z}+c=0and{L}_2:\overline{\beta }z+\beta \overline{z}+d=0$ where $\alpha =(a,b)and\beta =(p,q)$ are non –zero complex numbers. Then their cartesian equations are
$\begin{array}{l}ax+by+\frac{c}{2}=0andpx+qy+\frac{d}{2}=0\\ \therefore L_1\perp L_2\iff ap+bq=0\iff \alpha \overline{\beta }+\overline{\alpha }\beta =0\end{array}$ 
$\iff \frac{\alpha }{\overline{\alpha }}+\frac{\beta }{\beta }=0\iff {\mu }_1+{\mu }_2=0$ where ${\mu }_1=-\frac{\alpha }{\overline{\alpha }}and{\mu }_2=\frac{-\beta }{\beta }$ are the complex slopes of $L_1$ and $L_2$ respectively.
 $\begin{array}{l}{L}_1\to L_2\iff aq-bp=0\iff \alpha \overline{\beta }-\overline{\alpha }\beta =0\\ \iff \frac{\alpha }{\overline{\alpha }}=\frac{\beta }{\overline{\beta }}\iff {\mu }_1={\mu }_2\end{array}$