Find all the complex numbers $z$ satisfying 
Find all the complex numbers $z$ satisfying
  $$
  \bigg|\frac{1+z}{1-z}\bigg|=1
$$

So far I´ve done this:
$$
  z=a+bi \\
  \bigg|\frac{(1+a)+bi}{(1-a)-bi}\bigg|=1 \\
  \mathrm{expression*conjugated} \\
  \bigg|\frac{1+2bi-(a^2)-(b^2)}{1+2a+a^2+b^2}\bigg|=1
$$

Hint:  $|c/d|=|c|/|d|$ and your numerator and denominator then represent the distances from ??????
