If you want to quantify the shape of a peak, then the inverse of the Hessian seems more intuitive than the Hessian itself. E.g. for the PDF of a normal distribution, the inverse of the Hessian corresponds to the covariance matrix. But for the inverse Hessian, large determinant does mean broad basin, unlike for the standard Hessian. And the inverse Hessian has basically the same off-diagonal elements as the Hessian does.
I have a possible fix to the argument:
If you want to quantify the shape of a peak, then the inverse of the Hessian seems more intuitive than the Hessian itself. E.g. for the PDF of a normal distribution, the inverse of the Hessian corresponds to the covariance matrix. But for the inverse Hessian, large determinant does mean broad basin, unlike for the standard Hessian. And the inverse Hessian has basically the same off-diagonal elements as the Hessian does.