There are a couple of ways to do that. You can use hidden guide objects to keep track of rotation offsets and/or alignments. But I'd recommend some simpler approaches if you are using a static overhead perspective that doesn't shift around in terms of pitch (so you are looking down on your object from above on the Y plane all the time).
Lets say your camera has tilted to the left as in your example image above, maybe 45 degrees. You can keep track of the Y (yaw) axis rotation for the camera in a variable, then apply that as an offset to the object before moving it:
CamY#=45.0 // just a stored value for the camera's yaw axis
SetCameraRotation(1,90,CamY#,0) // looking down from above on Y
// now offset your object to that same rotation before moving
// start by capturing the objects current angles so you can set it back when finished
// this is optional though only if you need to preserve the original orientation
OldObjX#=GetObjectAngleX(Obj)
OldObjY#=GetObjectAngleY(Obj)
OldObjZ#=GetObjectAngleZ(Obj)
// now set the object to the camera's Y orientation
SetObjectRotation(Obj,0,CamY#,0)
// now when you move 'up' with the object, it moves up relative to the camera
MoveObjectLocalZ(Obj,Distance#)
// finally, restore the object back to its original orientation if needed
SetObjectRotation(Obj,OldObjX#,OldObjY#,OldObjZ#)
If you're not using variables to keep track of the camera's orientation, you can just align it directly:
// offset your object to that same rotation before moving
// start by capturing the objects current angles so you can set it back when finished
// this is optional though only if you need to preserve the original orientation
OldObjX#=GetObjectAngleX(Obj)
OldObjY#=GetObjectAngleY(Obj)
OldObjZ#=GetObjectAngleZ(Obj)
// now set the object to the camera's Y orientation
SetObjectRotation(Obj,0,GetCameraAngleY(1),0)
// now when you move 'up' with the object, it moves up relative to the camera
MoveObjectLocalZ(Obj,Distance#)
// finally, restore the object back to its original orientation if needed
SetObjectRotation(Obj,OldObjX#,OldObjY#,OldObjZ#)
