= Solution
Write $z=x+iy$. Rationalizing gives
$$
\frac{z-2}{z}
=\frac{x^2-2x+y^2+2iy}{x^2+y^2}.
$$
An argument of $\pi/4$ requires equal positive real and imaginary parts. Hence
$$
x^2-2x+y^2=2y,
\qquad y>0.
$$
Completing squares yields
$$
\boxed{(x-1)^2+(y-1)^2=2,\qquad y>0}.
$$
Thus the <constant-argument locus of a Möbius transformation> is the part above the real axis of the circle with centre $(1,1)$ and radius $\sqrt2$. Its endpoints $z=0$ and $z=2$ are excluded because the quotient is respectively undefined and zero.
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