Considering the algebra $A=\mathbb{C}[z;\sigma]$ of the skew polynomials, which is like $\mathbb{C}[z]$ but with the multiplication of elements defined the following way: $xb=\sigma(b)x$ and extending it to $\sigma^n(b)$: \begin{equation} \sigma^n(b)= \begin{cases} b & \text{if $n$ is even} \\ \overline{b} & \text{if $n$ is odd} \end{cases} \end{equation}
I have to write an $\mathbb{R}$-Algebra isomorphism \begin{equation} \mathbb{C}[z;\sigma]\cong \mathbb{R} \langle x,y\rangle/(y^2+1,xy+yx) \end{equation} but don't quite understand the second ring, I get that elements in $\mathbb{R} \langle x,y\rangle/(y^2+1,xy+yx)$ have a degree of at most $1$ in $y$, but don't really understand what $xy+yx$ does to the elements, and why is it not simply $2xy$ if I presume elements in that ring are commutative? Because of this, I can't seem to quite find the bases to form the isomorphism.