I was looking for a classification of quadrics in 4D and I found this thread from 9 years ago.

http://hddb.teamikaria.com/forum/viewtopic.php?f=3&t=352

It didn't go very far, but this could be really useful for PDEs and coordinate systems.

I'd really like to have all the quadrics in 4D listed and named. This Wiki article is a good place to start:

https://en.wikipedia.org/wiki/Quadric#Euclidean_plane_and_space

Here I've listed all the quadrics up to 4D with the constants removed.

Quadrics in 2D:

Ellipse: EE : x^2 + y^2 = 1

Hyperbola: EH : x^2 - y^2 = 1

Parabola: EP : x^2 - y= 0

Rectangle: II

Quadrics in 3D:

Ellipsoid: EEE : x^2 + y^2 + z^2 = 1

Hyperboloid with one sheet: EEH : x^2 + y^2 - w^2 = 1

Hyperboloid with two sheets: EHH: x^2 + y^2 - z^2 - w^2 = 1

Elliptic paraboloid: EEP: x^2 + y^2 - z = 0

Hyperbolic paraboloid: EHP: x^2 - y^2 - z = 0

Elliptic cylinder: EEI : x^2 + y^2 = 1

Hyperbolic cylinder: EHI : x^2 - y^2 = 1

Parabolic cylinder: EPI : x^2 - y= 0

Cuboid: III

I was tempted to name x^2 - y - z = 0, but this is actually just a rotated parabolic cylinder. So we'll never find a P^2. Also note that the equation for hyperboloid with two sheets is the negative of the equation in Wikipedia, to match the pattern better.

Quadrics in 4D:

EEEE: x^2 + y^2 + z^2 + w^2 = 1

EEEH: x^2 + y^2 + z^2 - w^2 = 1

EEHH: x^2 + y^2 - z^2 - w^2 = 1

EHHH: x^2 + y^2 - z^2 - w^2 = 1

EEEP : x^2 + y^2 + z^2 - w = 0

EEHP : x^2 + y^2 - z^2 - w = 0

EHHP : x^2 - y^2 - z^2 - w = 0

There are some simple cylinders:

EEEI

EEHI

EHHI

EEPI

EHPI

EEII

EHII

EPII

IIII

And finally some double cylinders:

EE x EE

EE x EH

EE x EP

EH x EH

EH x EP

EP x EP

It seems like in n dimensions there are n pure quadrics, n - 1 parabolic quadrics, and then a bunch of cylinders, which will relate to the number of rotatopes in some way.