Can a finitely generated commutative $\mathbb C$-algebra $A$ have (a) an indecomposible or (b) irreducible representation of dimension $>1$ (over $\mathbb C$)?
Without the finite generation condition, I believe the answer is Yes. For example, I believe the group algebra $A=\mathbb C\mathbb Q$ (for the additive group $\mathbb Q$) is an irreducibible representation of itself.
Based on Mariano's answers: (a) The regular representation of any integral domain $A$ is indecomposible. Indeed, if $A=I\oplus J$ as an $A$-module then $I\cdot J\in I\cap J=\{0\}$.
(b) Every irreducible representation $V$ of a finitely generated commutative $\mathbb C$-algebra $A$ is $1$-dimensional. Indeed, let $\mathfrak{m}$ be a maximal ideal containing the anihilator $\mathrm{Ann}(V)$ of $V$. Then $\mathrm{Ker}(V=A/\mathrm{Ann}(V)\to A/\mathfrak{m})$ is a proper submodule of $V$ and, hence, trivial. Consequently, $V=A/\mathfrak{m}$. By Zariski Lemma, the field $A/\mathfrak{m}$ must be a finite extention of $\mathbb{C}$ and, hence, $\mathbb{C}$.