A zero-dimensional local ring is Cohen--Macaulay

Every element is uniquely $a+b\epsilon$ with $a,b\in k$. It is a unit when $a\mathrm{\ne }0$, with inverse $a^{-1}-b{a}^{-2}\epsilon$; when $a=0$ it lies in $(\epsilon )$. Thus $(\epsilon )$ is the unique maximal ideal. Any nonzero ideal contained in $(\epsilon )$ contains $b\epsilon$ for some $b\mathrm{\ne }0$ and hence contains $\epsilon$, so the ideals are exactly $0$, $(\epsilon )$ and $A$. Consequently $A$ is Artinian and Noetherian, and it is nonreduced because $\epsilon \mathrm{\ne }0$ but ${\epsilon }^2=0$.

Every prime ideal contains the nilpotent element $\epsilon$, so the unique prime is $(\epsilon )$ and $\dim A=0$. Every element of $(\epsilon )$ annihilates the nonzero element $\epsilon$, hence no element of the maximal ideal is $A$-regular and $\mathrm{depth}A=0$. Therefore $A$ is Cohen--Macaulay by [F1].