= Solution
{c}
If $z\neq w$, apply translation, rotation and <Brownian scaling> to
$$
\widetilde Z_t=1+\frac{Z_{|z-w|^2t}-z}{z-w}
=\frac{Z_{|z-w|^2t}-w}{z-w}.
$$
This is standard <planar Brownian motion> started at one. Its almost-sure visits to arbitrarily small discs about zero translate to visits of $Z$ to discs about $w$, with radii multiplied by $|z-w|$. Therefore \b[the same neighborhood recurrence holds for every pair $z,w$]. If $z=w$, every such <disc> is already hit at time zero. If positive hitting times are requested, <continuity> gives positive times within every fixed <disc> as well. A countable sequence of shrinking radii suffices to obtain one probability-one event for this assertion.
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