In wikipedia (link), it says:
(i) Therefore, a harmonic function $u$ admits a conjugated harmonic function if and only if the holomorphic function $g(z)\colon=u_{x}(x,y)-iu_{y}(x,y)$ has a primitive $f(z)$ in $\Omega$, in which case a conjugate of $u$ is, of course, $Imf(x+iy)$.
(ii) So any harmonic function always admits a conjugate function whenever its domain is simply connected, and in any case it admits a conjugate locally at any point of its domain.
Actually, I can understand the statement (i). But I cannot prove (ii) and I'm looking for a rigorous proof of (ii).
Consider the differential form over $\Omega\subset\mathbb{R}^2$ $$\omega:=-(\partial_y u)dx + (\partial_x u)dy$$ since it's $C^1$ and closed (because $u$ is harmonic), if we have that $\Omega$ is simply connected, then $\omega$ has a primitive (by Poincarè Lemma) $v:\Omega \rightarrow \mathbb{C}$, which is the harmonic conjugate of $u$ (check derivatives).