Where did I go wrong? 
I'm trying to compute the harmonic conjugate of $u(x,y)=x-y$ on $\mathbb R^2$

Let $v$ be the harmonic conjugate of $u(x,y).$ Then $u_x=v_y\implies v_y=-1\implies v=y+\phi(x).$
Also, $u_y=-v_x\implies 1=\phi'(x)\implies\phi(x)=x+d\implies v=y+x+d$ 
Is it correct?

Note that from $$u_x=1$$ $$u_y=-1$$ you get that $$v_x=1$$ $$v_y=1$$
It follows that $v(x,y)=y+C(x)$ and $v(x,y)=x+C_1(y)$ for each $y$, for some unknown $C,C_1$.
Subtract. What do you get? 
SPOILER

 Then for every $x,y$, $$0=y-x+C(x)-C(y)$$ This means that $C(x)=x$ and $C(y)=y$.

