Unit decomposition of the 2-adic Gaussian field (${\mathbb{Z}}_2[i{]}^{\times }={\mu }_4\times (1+(1-i{)}^3{\mathbb{Z}}_2[i])$)

The Eisenstein polynomial of $1-i$ is $T^2-2T+2$, so ${\mathbb{Q}}_2(i)$ has residue field ${\mathbb{F}}_2$ and ramification index two. The four roots $1,i,-1,-i$ have distinct cosets modulo $U_3$, which has index four by the successive quotients of principal-unit groups. The logarithm isomorphism on deep principal units gives $U_3\cong ({\mathbb{Z}}_2[i],+)$. Hence these four roots split the full unit group, and are all its torsion.