= 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 $N\ne0$. On <restriction of scalars>,
$$
Y=T^2=\lambda^2I+2\lambda N.
$$
For $\lambda=0$ this is the zero operator on a two-dimensional space, which splits as two one-dimensional <submodules>. For $\lambda\ne0$, its <nilpotent> part is nonzero of square zero, so it is one <Jordan block> and is indecomposable. Therefore the full list is
$$
\boxed{A/((X-\lambda)^2),\qquad\lambda\in\mathbb C\setminus\{0\}.}
$$
Different $\lambda$ give nonisomorphic $A$-<modules>; the restricted $B$-<modules> for $\lambda$ and $-\lambda$ are isomorphic.
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