Denote $z = (a, b)$ and $w = (c, d)$. Using the definitions
we have
\begin{equation}
\begin{split}
\overline{z w}
= \overline{(a, b) (c, d)}
& = \overline{(a c - b d, a d + b c)} \\
& = (a c - b d, - a d -b c) \\
& = (a, -b)(c, -d) \\
& = \overline{(a, b)} \overline{(c, d)} \\
& = \overline{z} \, \overline{w}
\end{split}
\end{equation}
$\square$
