For any non-zero complex number

For complex number z

Say $z=x+i y=r e^{i \theta}$ where $r=$ modulus of $z, \theta=$ argument of $\bar{z}=x-i y$

$\Rightarrow \bar{z}=r e^{-i \theta}$

Let us arg z = θ

Since $\arg \bar{z}=-\arg z$

$\Rightarrow \arg z+\arg \bar{z}=\theta+(-\theta)$

i.e $\arg z+\arg (\bar{z})=0$