= Solution
\b[No such <conformal bijection> exists.] A prescribed point is a <polar point for planar Brownian motion>: a <planar Brownian motion> started away from zero hits zero with probability zero. For example, inside an annulus the probability of reaching radius $\varepsilon$ before radius $R$ is
$$
\frac{\log R-\log|z'|}{\log R-\log\varepsilon},
$$
by the <harmonic function> $\log|w|$ and the <optional stopping theorem>. For fixed $R>|z'|$ this tends to zero as $\varepsilon\downarrow0$. Any finite-time visit to zero would occur before leaving some sufficiently large disc; the countable union over integer $R$ still has probability zero.
The exit time from $\mathbb C\setminus\{0\}$ is therefore infinite <almost surely>, contradicting part (b) if this were the image of a bounded domain. Equivalently, a hypothetical inverse $\psi:\mathbb C\setminus\{0\}\to D$ would be bounded and holomorphic. The <Riemann removable singularity theorem> extends it over zero, and the <Liouville theorem> makes it constant, contradicting bijectivity.
$$
\boxed{D\text{ bounded}\quad\Longrightarrow\quad
D\text{ is not conformally equivalent to }\mathbb C\setminus\{0\}.}
$$
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