$\newcommand{\iu}{{i\mkern1mu}}$Gaussian integers are complex numbers $a + b \iu$ where $a$ and $b$ are integers, and $\iu^2 = -1$.
Eisenstein integers are complex numbers $a + b \omega$ where $\omega= e^{2 \pi i/3}$, so that $\omega^3 = 1$, i.e., $\omega$ is a cube-root of unity.
Q. Are there natural generalizations of these communtative rings to $a + b \gamma$, where $a$ and $b$ are integers, and $\gamma$ is an $n$-th root of $\pm 1$?
Do they have names? Applications?
The set of complex numbers of the form $a + b \gamma$, where $a$ and $b$ are integers, form a ring iff $\gamma$ is a quadratic algebraic number.
Now, $\gamma$ is an $n$-th root of unity and a quadratic algebraic number iff $n \in \{3,4,6\}$.
Thus, there is only one possible additional example: $\gamma=e^{2 \pi i/6}$, a primitive $6$-th root of unity. However, $-\gamma$ is a primitive cubic root of unity and so this ring is the same as the Eisenstein integers.
Therefore, your examples are the only possible ones.