 Solution

A two-dimensional indecomposable module over $\mathbb{C}[X]$ is exactly $A/((X-\lambda {)}^2)$ for some $\lambda \in \mathbb{C}$. Write its operator as $T=\lambda I+N$ with $N^2=0$ and $N\ne 0$. On restriction of scalars,For $\lambda =0$ this is the zero operator on a two-dimensional space, which splits as two one-dimensional submodules. For $\lambda \ne 0$, its nilpotent part is nonzero of square zero, so it is one Jordan block and is indecomposable. Therefore the full list isDifferent $\lambda$ give nonisomorphic $A$-modules; the restricted $B$-modules for $\lambda$ and $-\lambda$ are isomorphic.