"turning on the Z axis" (dunno if i said it correctly, could be also X or Y) would essentially be changing the plane, meaning, the arena would be a cube instead of a 2 dimensional plane. Let's say a couple of players would create a dead end you couldn't avoid by going up, down, left or right, you could rotate the view.

For example on a normal snake 2D game, pressing the key oposite and to the direction your'e heading is meaningless, you can't go straight back and your'e already going forward. On a cube those keys could be mapped to 3D rotation.

You would still see the game as 2D and depth would be emulated for example with transparency, were close trails are opaque and deeper objects are transparent, so that you could know if you could turn or not, for example on a trail right in front of you.

edit: fix typos

ps.: of course playing on a cube is only be interesting when you have a lot of players in the game, or else it would be very impractical to corner someone (since you would need to get him wrapped completely)

How would you ever find yourself on the same plane as the other players? Or even on any plane that they had ever been on? If the trails are infinitely thin lines arbitrarily positioned in 3D space then it's effectively impossible to hit them.

You could quantize them on some or all of the axes but I would find that less elegant than keeping the game in continuous space and making the trails planar somehow.

no, because everyone's moving in x,y,x integer coordinates limited to the cube's dimension, max x,y,x = 1000 for example, and the server records the coordinates you'be moving along (0,0,1), (1,0,0), (2,0,0). when you move into a coordinate someone (or you) has been, its a collision. although you move 1 point at a time, the line thickness in the Ui may be more than 1 pixel, it's just on the UI level.

you start on the usual plain:

  y
  |
  | *-->
  |
  z-------x

y is the vertical coordinate and x is the horizontal. that's the plane your're seeing on your game client. if you're moving right along the x line, you're increasing the x coordinate (x,y, z) while the others are constant

if you press 'D' or 'right', you "turn" the view on the y axis, and then the plane you're on becomes:

  y
  |
  |  *-->
  |
  x-----z

now, y is the (same) vertical but the horizontal has become z, you keep moving right but x value has fixed and now z is increasing. if you press up, you start increasing y, but x, and z are still fixed (since you havent moved on their axis)

the server records the x,y,x coordinates you been through (for everyone) and when someone moves into a recorded coordinate, game over man.