Four colour theorem in $3$ dimensions?

If you take a set of square columns in the z=0 plane, such that there is a set for y=0, y=1, etc. Now take a second set, turned at right angles and at z=1, so you have x=0, x=1, &c. Now join these by pairs at x=n, y=n.

You now have an infinite number of X-shaped figures, each pair neighbouring another exactly twice (ie at i,j and j,i).

So there exists a construction fot infinite colours.

There are meaningful generalizations, if you consider surfaces like sphere, torus, Möbius band etc. as "3D objects". The minimum number of required colors for the mentioned surfaces is 4, 7 and 6, respectively.