I wanted to report on a variation, *Gaussian prime random walks*.
As before
start with any point in the complex plane $c_0$. Walk in the $+x$ direction
until you hit a [Gaussian prime](https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_primes).
Unlike the original question, which requires a $90^\circ$ counterclockwise turn,
turn left or right by $90^\circ$ with equal probability. Continue turning randomly $\pm 90^\circ$ at every Gaussian prime hit, until some Gaussian prime is hit for a
second time. Call that a *Gaussian prime loop*.

Below is an example:
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[![Loop62][1]][1]
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<sup>
Green: start at $c=100 + 2i$. Red: looped at $197 + 22 i$. Yellow: Gaussian prime turn.
</sup>
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Although this
runs into the same open-problem impediment
cited by @FrançoisBrunault,<sup>*</sup>
it appears that, for a random starting $c_0$ within some fixed distance of the origin,
the probability of eventually looping might be $1$.
Looping is much "easier" to achieve than is spiral cycling, which requires
revisiting $c_0$ from the left,
whereas looping just needs to revisit some prime twice.

<hr />
><sup>*</sup>
<sub>"It's unknown whether there are infinitely many Gaussian primes of the form $n+i$ with $n \in \mathbb{Z}$. So starting at $N+i$ and moving $+x$, we cannot exclude the possibility of hitting no prime."
</sub>



  [1]: https://i.sstatic.net/05Qjp.jpg