What is $f(x, y) = |x| - |y|$ called?
$f(x,y)=x^2-y^2$ is your friendly neighbourhood hyperbolic paraboloid. $f(x, y) = |x| - |y|$ naturally has similar appearance. Do shapes of the latter form have a name?
What about hyperbolic superparaboloid analogous to quadric - superquadric?