Is $\langle x^2+1,y\rangle$ maximal or prime in $\Bbb{R}[x,y]$ or $\Bbb{C}[x,y]$

The ideal $(x^2+1, y)$ is maximal in $\mathbb{R}[x, y]$, since $\mathbb{R}[x, y] / (x^2+1, y) \simeq \mathbb{R}[x] / (x^2 +1)$ by the third isomorphism theorem, and $\mathbb{R}[x] / (x^2 +1)$ is $\mathbb{C}.$
On the other hand, it is not a maximal ideal in $\mathbb{C}[x, y]$ (it is not even prime, in fact). This is because $\mathbb{C}[x, y] / (x^2+1, y) \simeq \mathbb{C}[x] / (x^2+1) \simeq \mathbb{C}[x] / (x - i) \times \mathbb{C}[x] / (x + i) \simeq \mathbb{C}^2$.