In polar form,
$z_1=r_1\mathrm e^{\mathrm i\phi_1}\;,$ $z_2=r_2\mathrm e^{\mathrm i\phi_2}\;,$
where $r_1$ and $r_2$ are the magnitudes and $\phi_1$ and $\phi_2$ are the arguments of $z_1$ and $z_2$, respectively, the multiplication takes the simple form
$z_1z_2=r_1r_2\mathrm e^{\mathrm i(\phi_1+\phi_2)}\;.$
Now you can read off the argument of $z_1z_2$ from the exponential, but you have to take into account that this might wrap around. The usual choice for the range of the argument is $(-\pi,\pi]$ (this is what's called the "principal argument"), so $\phi_1+\phi_2\in(2\pi,2\pi]$. Thus you may have to add $2\pi$ or subtract $2\pi$ to get this back into the range $(-\pi,\pi]$, so the difference between the argument of $z_1z_2$ and the sum $\phi_1+\phi_2$ of the individual arguments can be $-2\pi$, $0$ or $2\pi$.