2015 AMC 10A Problem 12

Substitute $x=\sqrt{\pi}$. Then $x^2=\pi$ and $x^4=\pi^2$, so the equation becomes $$y^2-2\pi y+\pi^2=1.$$ Hence $(y-\pi)^2=1$, giving the two possible values $y=\pi+1$ and $y=\pi-1$. Their distance is $2$. Thus, C is the correct answer.