Proving Two Complexes are Not Quasi-Isomorphic

As was pointed out by Andrea in the comments to the other answer, the other answer does not answer the question properly.

An argument that the two complexes cannot be quasi-isomorphic in the general sense is sketched on page 2 of:

https://www.math.upenn.edu/~tpantev/rtg09bc/secnotes/ainfnity.davidovich.pdf

I guess we are working with $\Bbb C[x,y]$-modules.

The degree-zero part of the quasi-isomorphism would have to be a morphism $\Bbb C[x,y] \oplus \Bbb C[x,y] \to \Bbb C[x,y]$ inducing an isomorphism between $K$ and $\Bbb C[x,y]$, where $K$ is the kernel of the differential of $C^*$, that is $$ K = \{ (y R, x R) \, | \, R \in \Bbb C[x,y]\}.$$

But such a morphism is necessary of the form $(P,Q) \mapsto PU + QV$, so it would map $K$ into the ideal $x \Bbb C[x,y] + y \Bbb C[x,y] \varsubsetneq \Bbb C[x,y]$, a contradiction.