for an exam:
given that |z|=|w|
by vector addition a rhombus will be formed (here you draw a diagram including both z+w and z-w)
therefore, the angle between diagonals is 90 degrees.
i.e. arg[(z+w)/(z-w)]= 90 degrees (using angle between two vectors)
hence, (z+w)/(z-w) is purely imaginary...