Example of a non-Noetherian complete local ring

I think that you should look at the ring $k[[X_1,X_2,...]]$ of power series with an infinity of variables. This ring is complete (as the completion of the ring $k[X_1,X_2,...]$ of polynomials with an infinity of variables) and its unique maximal ideal is the one consisting of power series without constant term. It is clearly not noetherian, considering the chain of ideals generated by $X_1$, $\{X_1,X_2\}$, $\{X_1,X_2,X_3\}$, etc.

Take an algebraic closure of $\mathbb{Q}_p$, the field of $p$-adic numbers, and complete it to get a complete field $\mathbb{C}_p$. It turns out that $\mathbb{C}_p$ is still algebraically closed. The valuation ring $R$ of $\mathbb{C}_p$ is a complete local ring whose maximal ideal $\mathfrak{m}$ satisfies $\mathfrak{m}^2=\mathfrak{m}$. It follows from Nakyama's lemma and the fact that $\mathfrak{m}\neq 0$ that $R$ is not Noetherian.

EDIT: I was thinking about this example recently and realized that it is not an example of the type requested by the OP, because the valuation ring $R$, while complete with respect to its valuation topology, is not $\mathfrak{m}$-adically complete due to the relation $\mathfrak{m}^2=\mathfrak{m}\neq 0$, which implies $\mathfrak{m}^n=\mathfrak{m}$ for all $n\geq 1$. This relation shows that $R$ is not $\mathfrak{m}$-adically separated, which is part of the definition of $\mathfrak{m}$-adically complete. Its $\mathfrak{m}$-adic completion is the residue field $R/\mathfrak{m}$, which is an algebraic closure of $\mathbf{F}_p$.