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 + c, - b - d) \\
& = (a, - b) + (c, - d) \\
& = \overline{(a, b)} + \overline{(c, d)} \\
& = \overline{z} + \overline{w}
\end{split}
\end{equation}
$\square$
