= Solution
{c}
Start a <planar Brownian motion> $B$ at zero and apply part (a) to the entire function $f(z)=e^z$. Then
$$
e^{B_t}=1+\widetilde Z_{A_t},\qquad
A_t=\int_0^t e^{2\operatorname{Re}B_s}\,ds.
$$
The left side never vanishes. The clock is continuous, starts at zero, and tends to infinity almost surely, so its range is all nonnegative times. Consequently the complex Brownian motion $1+\widetilde Z_s$ never reaches zero at any time $s$. Every <planar Brownian motion> started at one has that same law, hence
$$
\boxed{P(T<\infty)=0.}
$$
This is the <complex exponential construction of planar Brownian motion> and proves that a fixed point is a <polar point for planar Brownian motion>. The divergence of the clock is essential: it makes this a conclusion about all Brownian times.
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