If z=x+iy and w=1−izz−i, then |w|=1 implies, that, in the complex plane

|w|=1 ⇒ 1−izz−i=1⇒ −i2−iz=|z−i|⇒ |z+i|=|z−i| ⇒ |(−i)(z+i)|=|z−i|⇒z lies on the perpendicular bisector of the segment join-ing i and -i⇒z lies on the real axis.