Let z and w be two complex numbers such that $\mathrm{Re}\left(z\right)=|z-2|, \mathrm{Re}\left(w\right)=|w-2|$ and  $\arg (z-w)=\frac{\pi }{3}$, then  $\mathrm{Im}(w+z)$ is 

$\mathrm{Re}\left(z\right)=|z-2|\Rightarrow x=\sqrt{{(x-2)}^2+y^2} \Rightarrow y^2=4(x-1)$

$\therefore z=(1+t^2)+2ti$ (parametric form)
Similarly, $w=(1+s^2)+2si$
$$\begin{gathered} z-w=(t^2-s^2)+2i(t-s) \\ \Rightarrow \arg (z-w)=\frac{\pi }{3} \\ \Rightarrow {\tan }^{-1}\left(\frac{2}{t+s}\right)=\frac{\pi }{3} \Rightarrow \frac{2}{t+s}=\sqrt{3} \\ \Rightarrow t+s=\frac{2}{\sqrt{3}} \\ Now, z+w=2+t^2+s^2+2i(t+s) \\ \therefore I_m(z+w)=2(t+s)=\frac{4}{\sqrt{3}} \end{gathered}$$