Is the ideal $(x-y,x+y)$ same as $(x,y)$, since $$x+y, x-y \in \mathbb{C},$$ so $ y \in \mathbb{C} $ because $\mathbb{C}$ is a field. And similarly $x $ in $\mathbb{C}$, so $ (x,y)\in (x-y,x+y)$ and the converse follows similarly.
This would show that $(x,y)$ is maximal in $\mathbb{C}[X,Y]$, which is what I originally wanted to prove.
Thank you
I've failed in understanding about ideals in which ring is this topic, so my answer is about $\mathbb C[x, y]$.
The easiest way to prove that$(x-y, x+y) = (x, y)$ is to prove that $x, y \in (x-y, x+y)$ and $x - y, x + y \in (x, y)$. Both of these facts are trivial (for example, $y = \frac{1}{2}((x + y) - (x -y)) \in (x-y, x +y))$.