Character movement angled aerial 2D perspective

Hi guys,

I'm about to create a game that will have an angled 2D perspective, kind of like the perspective in this screenshot: http://bit.ly/ItB9zM. The player will be able to move sideways, up, down and diagonally as well as being able to jump, which is causing me trouble.

The ground sprite that he is walking on is positioned in the y axis, like the background in the Unity 2D tutorial, meaning that gravity needs to be disabled as the character does not have any colliders beneath him to rest on. Therefore I cannot check if the player is grounded and because I've got no gravity I cannot comprehend how I should program a parabola-like jumping.

The idea is that there should be a shadow beneath the player which stays in place when the jumping occurs. Also the player should have the possiblity to move up and down while jumping without affecting the jump height. Where should I start in such implementation?

I have heard something about setting the z-axis equal the the y-axis, but I do not understand how this should help me.

Another example of the perspective can be seen here (the way he uses the jetpack is kind of what I had in mind for the jumping, apart from ability to steer the player) : http://youtu.be/bmAbUs-P7eY?t=27s

Any help as to how I can achieve this is very much appreciated.