 Critical points escaping to infinity in a Gaussian-weighted bilinear function

The origin is a saddle point of a scalar function for every real parameter. For $\alpha >0$ four further critical points have $x^2=y^2=1/(2\alpha )$. Equal-sign coordinates give global maxima and opposite-sign coordinates give global minima, with values $\pm 1/(2e\alpha )$. The Hessian matrix there is $-4\alpha xy{e}^{-1}I$. Their distance from the origin is ${\alpha }^{-1/2}$, so as $\alpha \downarrow 0$ they escape every compact region. At nonpositive $\alpha$ the origin is the only critical point. This illustrates why the number of critical points on an unbounded domain can change without a finite degenerate critical point appearing.