We have investigated properties of the positional numeration system with $β = i - 1, 𝒟 = {0, \pm 1, \pm i}$ . We proved that every Gaussian integer has a unique $3$ -NAF representation in this system, which is always optimal and an average digit of such a representation has only a $\frac{1}{4}$ probability to be non-zero. We have also calculated the maximum number of optimal representations a Gaussian integer can have if its $3$ -NAF representation has exactly $N$ non-zero digits, as well as which exact Gaussian integers achieve this maximum. The result is similar to that in [num-binary-signed-reprs], except with a different recurrent sequence, given by

r_{- 1} ≔ 3, r_{0} ≔ 8, r_{1} ≔ 17, r_{N + 3} ≔ r_{N + 2} + 2 r_{N + 1} + 2 r_{N},

with the maximum number of optimal representations for a given $N$ being $r_{N - 3}$ .