I have no ideas about a hypercone. Never thought about it, and not sure how to start. Here is a try.

Equations of a cone. Note: First 2 are equations of a circle in 2D (If R is constant).
X= R*cos(A)
Y = R*Sin(A)
Z=R*Constant
In the above, R & A are variables.

Equations of hypercone? First 3 are equations of a sphere in 3D (If R is constant).
X1=R*cos(A)*cos(B)
X2=R*cos(A)*sin(B)
X3=R*Sin(A)
X4=R*Constant
In the above, R, A, & B are variables. Thinking of A as latitude and B as longitude helps visualize the sphere.

I am pretty sure that the above are correct, but it does not help me much in visualizing it.

Instead of all points on a circle connected by lines to a point, maybe all points on a sphere connected by lines to a point without any line intersecting the sphere? I do not like this.

Instead of a succession of ever larger circles connected to each other, a succession of ever larger spheres connected to each other. This seems correct, but I cannot visualize it. Imagine just two of the circles from an ordinary cone. Every pair of corresponding points must be connected. Now consider two spheres with every pair of corresponding points similarly connected by lines which do not intersect any of the other points on the two spheres. I cannot do it.

Instead of a 2D object (triangle) twisted in 3D space so that two sides coincide, a 3D object (??) twisted in 4D space so that two Planes (??) coincide? This does not seem right, even if I could decide what 3D object to twist. All I get from this is one cone inside another from twisting a solid triangle.

The hypercylinder is not much different from the hypercone. For a cylinder, the circles are the same size instead of being different sizes like the cone. For a hypercylinder, the spheres are all the same size.