One of my favourite open problems in number theory, an area in which I enjoy only as a hobbyist, is the Gaussian moat problem, namely

"Is it possible to walk to infinity in $\mathbb{C}$, taking steps of bounded length, using the Gaussian primes as stepping stones?"

We can easily show that one cannot accomplish walking to infinity using steps of bounded length on the real line using primes in $\mathbb{R}$. For an arbitrary natural number $k$, consider the $k-1$ consecutive numbers

$$ k! + 2, k! + 3, \ldots k! + k, $$ all of which are composite. This is another way of saying there are arbitrarily large gaps in the primes.

For the Gaussian primes, there is computational proof that a moat of length $\sqrt{26}$ exists, so one cannot walk to infinity using steps of length $5$. Erdos is said to have conjectured that it is impossible to complete the walk. Percolation theory also suggests that the walk is impossible, though to my understanding this heuristic assumes the primes are completely independent in some way.

Eisenstein integers are numbers of the form $a+b\omega$, with $a$, $b \in \mathbb{R}$, where $\omega = \mathrm{e}^{\mathrm{i}\pi/3}$. My first and main question is -

What is the current lower bound for step size in the analogous problem for Eisenstein primes?

Quaternions with all integer components are called Lipshitz integers. So let us call primes over this ring Lipshitz primes. A Lipshitz integer is only a Lipshitz prime if its norm is a prime. Is anything known about the moat problem over $\mathbb{H}$? One might think that given the extra dimensions or degrees of freedom walking to infinity should be easier, however I'm not sure how rare Lipshitz primes are.

Responses to this post point out that factorisation over octonions is not unique, so it is difficult to come up with a concept of primes over $\mathbb{O}$.