 Solution

If $z\ne w$, apply translation, rotation and Brownian scaling toThis 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 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.