$ Re(z)+Im(z)= \frac{1}{Re(z)-Im(z)}$
Replace Re(z) with x and Im(z) with y to get:
$ x+y= \frac{1}{x-y}$
Then: $ (x+y)(x-y)=1 $
$ x^2-y^2 =1 $
Assuming i did my math properly all i have to do is sketch the region of $x^2-y^2 =1 $ . Help is appreciated. Thank you