$u,v$ are harmonic conjugate with each other in some domain $u,v$ are harmonic conjugate with each other in some domain , then we need to show 
$u,v$ must be constant.
as $v$ is harmonic conjugate of $u$ so $f=u+iv$ is analytic.
as $u$ is harmonic conjugate of $v$ so $g=v+iu$ is analytic.
$f-ig=2u$ and $f+ig=2iv$ are analytic, but from here how to conclude that $u,v$ are constant? well I know they are real valued function, so by open mapping theorem they are constant? 

Your proof is correct. I add some remarks: 


*

*$v$ is a conjugate of $u$ if and only if $-u$ is a conjugate of $v$ (since $u+iv$  and $v-iu$ are constant multiples of each other)

*Since the harmonic conjugate is unique up to additive constant, the assumption that $u$ is a conjugate of $v$ implies (because of 1) that $u=-u+\text{const}$, and conclusion follows.

*Related to 1: the Hilbert transform $H$ satisfies $H\circ H=-\text{id}$.

