We know for any principal ideal domain, objects in the bounded derived category are all formal hence we can classify those objects with finitely generated cohomology using structure theorem for finitely generated modules. 


Now consider the bounded derived category of $\mathbb C[x]/x^2$-modules, how to classify indecomposable objects with finitely generated cohomology in this category? Examples include $\mathbb C[x]/x^2 \overset{x}{\rightarrow} \mathbb C[x]/x^2 \overset{x}{\rightarrow}... \overset{x}{\rightarrow} \mathbb C[x]/x^2 $ and $\mathbb C$.


Note there exists non-formal object. 



Indecomposable objects in bounded derived category of $\mathbb C[x]/x^2$-mod
Up to shifts, every indecomposable object is of one of the forms described in the question.


I don't know an explicit reference, but here's a sketch of a proof.


By induction on the length, it's not hard to prove that every bounded complex of finite rank free modules, such that the image of each differential is contained in the radical of its codomain, is a direct sum of complexes of the form
$\mathbb C[x]/x^2 \overset{x}{\rightarrow} \mathbb C[x]/x^2 \overset{x}{\rightarrow}... \overset{x}{\rightarrow} \mathbb C[x]/x^2 $.


Now consider a minimal projective resolution of an object of the bounded derived category, truncated to the left of its homology.


By the way, this is an example of a "derived discrete" algebra, and there's a fair amount of literature about these.


