Let the complex number be written in polar form as $$z = e^{i\theta}$$ because $$|z| = 1$$. The only banned value is $$z \neq -1$$, i.e. $$\theta \neq \pi$$, which keeps the denominator in $$w$$ non-zero.
Define $$w$$ as given:
$$w = \frac{z - 1}{z + 1}$$
We will compare $$w$$ with its complex conjugate $$\bar w$$ to find the locus.
Since $$|z| = 1$$, we have the important identity $$\bar z = \frac{1}{z}$$.
Compute the conjugate of $$w$$:
$$
\bar w = \frac{\bar z - 1}{\bar z + 1}
= \frac{\tfrac{1}{z} - 1}{\tfrac{1}{z} + 1}
= \frac{1 - z}{1 + z}
= -\frac{z - 1}{z + 1}
= -w.
$$
Thus we have the relation
$$\bar w = -\,w \quad -(1)$$
Let $$w = x + iy$$ where $$x, y \in \mathbb{R}$$.
Then $$\bar w = x - iy$$.
Substituting into equation $$(1)$$ gives
$$x - iy = -x - iy.$$
Equate the real parts: $$x = -x \; \Longrightarrow \; x = 0.$$
The imaginary parts already match automatically.
Hence every admissible $$w$$ has zero real part and an arbitrary imaginary part:
$$w = 0 + iy.$$
This is precisely the entire imaginary axis in the Argand plane (excluding no additional points because $$z \neq -1$$ only removes a single value of $$w$$).
Therefore, the required locus is the imaginary axis.
Option B which is: the imaginary axis.
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