= Solution
Factor $e^{i\phi/2}$ from the numerator and denominator:
$$
w=\frac{e^{i\phi}-1}{e^{i\phi}+1}=\frac{2ie^{i\phi/2}\sin(\phi/2)}{2e^{i\phi/2}\cos(\phi/2)}=i\tan\frac\phi2.
$$
For $0<\phi<\pi$, the tangent ranges over all positive real values. Thus $\boxed{\operatorname{Re}w=0,\quad\operatorname{Im}w>0}$. Using the principal <complex logarithm>,
$$
\boxed{\log w=\ln\tan(\phi/2)+\frac{i\pi}2.}
$$
The real part ranges over the entire real line, while the imaginary part is fixed: the locus is the horizontal line $\operatorname{Im}\log w=\pi/2$. Any other fixed <logarithm branch> gives the parallel line $\operatorname{Im}\log w=\pi/2+2\pi k$ for one integer $k$; allowing every branch gives the union of those lines.
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