Prove that $\mathbb{R}[x]/(x^2+1)^2$ is isomorphic to $\mathbb{C}[y]/(y^2)$ as a $\mathbb{C}$-algebra.

Question

How could I prove that $A=\mathbb{R}[x]/(x^2+1)^2$ is isomorphic to $\mathbb{C}[y]/(y^2)$ as a $\mathbb{C}$ algebra? Are there any known isomorphisms or is there a trivial way to do so? I have been able to identify A as $\mathbb{R}[x]_{\leq 3}$ but don't really know how to construct the isomorphism. Also, does it affect in anything the fact that it is as a $\mathbb{C}$-algebra?