I wonder if Knight's Tours have been explored in higher dimensions, using the following definition of a knight move.

In dimension $d=2$, the knight moves left/right and forward/back one step and two steps in either order. This leads to $8$ moves.

In dimension $d=3$, the knight moves $\{ \pm 1, \pm 2, \pm 3 \}$ steps in any order along $x,y,z$ axes. This leads to $48$ moves; see below.

In dimension $d$, the knight moves $\{ \pm 1, \pm 2, \ldots \pm d \}$ steps in any order along the axes. This leads to $d!\, 2^d$ moves.

I have found literature on knight's tours in 3D but under a different (planar)
definition of move.^{1}

It would be interesting to learn which (if any!) $n \times n \times n$ cube chess boards allow a knight's tour.

^{1}Awani Kumar. "Magic Knight's Tours in Higher Dimensions." (arXiv link)

**Addendum**.

*BenBarber*found much more relevant references for planar knight moves in higher dimensions: "with $n$ even, there is always a knight's tour provided that $n$ is sufficiently large."

*TMA*made an incisive observation that answered my explicit question: There is no knight's tour for $d=3,4$, due to parity considerations. So, what remains is: (a) $d \ge 5$ and other odd triangular numbers, or (b) knight's tours of the even-coordinate-sum ("black") cells in $d=3,4$.