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Write $z=x+iy$. Rationalizing givesAn argument of $\pi /4$ requires equal positive real and imaginary parts. HenceCompleting squares yieldsThus the constant-argument locus of a Möbius transformation is the part above the real axis of the circle with centre $(1,1)$ and radius $\sqrt{2}$. Its endpoints $z=0$ and $z=2$ are excluded because the quotient is respectively undefined and zero.