Thus in the fourth dimension, there must be a third component of energy waves.
I suggested "tetric field" in an earlier thread, but I never really liked that name. Anyway, I think it's more sensible to represent the 4D magnetic field by a 2D tensor field, instead of having two 1D vector fields.
In nD, we can use either an (n-2) tensor field (the "classical" magnetic field), or it's duel, which is always a 2D tensor field (a bi-vector). The two fields are always perpendicular.
Since very few people here know how to work with tensors, it might be possible to describe them using a pair of vectors. But that's not the same as having three separate fields.
I haven't thought about EM waves using tensors, but I guess it would be pretty simple. You have the electric field oscillating in one direction, and the (classical) magnetic field perpendicular to the electric field and the direction of motion. I do know that Maxwells equations are the same in any dimension (if you replace the cross product with ^*).