I guess there must standard examples but I can't seem to find them.
I tried ${A = \mathbb{R}[x,y]/(x^2 -1, y^2 x + x)}$, which is not the domain of a homomorphism to $\mathbb{R}$ because it would have to send $x$ to $1$, and so $y^2$ to $i$.
But then I have trouble checking that there is no homomorphism ${\mathbb{C} \rightarrow A}$.
So I tried ${\mathbb{R}[x,y]/(x^2 -1, y^2 x + x, y^3 + y)}$ hoping it would be easier but no luck.
Do you know a simple example?