The rules and the examples (with diagrams) given in the PHB only focus on a battle grid layout. They don't specifically mention any vertical aspect.
They don't specifically mention any vertical aspect. They do specifically mention how the powers work, in terms that function perfectly fine for vertical dimension, without any indication we should suddenly stop following what the rules rather clearly say. And while the diagrims show only horizontal instances, there is nothing in the rules that is similarly restricted.
Yes, you could extrapolate that it goes out in all directions, I'm just saying that there's no guidance on this.
Sure, but the way the rules are written does not in any way imply a limitation to a single horizontal plane. Yes, it would be nice for them to have outright discussed it, but your interpretation requires specifically adding in changes to how they describe the rules working.
Also, does it make a lot of sense for the Fighter powers to also have a vertical component?
Well... yes. I would certainly say a fighter who is attacking all enemies in a Close Burst 1 - which is to say, swinging his sword at every enemy in reach - would be allowed to attack both an enemy in front of him, as well as an enemy directly over his head. I would, in fact, be quite appalled at a DM who ruled otherwise.
Or take my Dragonborn. I took Englarged Breath Weapon, which makes it a Close Blast 5. According to your rules, that would be 5 squares of vertical as well, meaning an effective volume of 125 squares! Doesn't that seem just a bit overpowered for a close blast weapon?
Not at all, since I don't expect to fight a bunch of soldiers who suddenly form a human pyramid. There circumstances in which that would be abusable are essentially non-existent. And if there
is a bunch of bats, all flying at different heights, but all within a few squares of each other, I would find it extremely underpowered for a DM to start claiming the wizard's fireball can only hit one of them.
All I'm saying is that in this example you have the dragon on one plane, PC's on another. As such, I would keep the fight simple and treat all effects as taking place on one plane a they do the other 99.9% of the time that you're playing.
That doesn't make things simple at all - it raises all sorts of questions. In fact, if you are forcing the dragon to only cast his Lightning Burst down the relative plane between himself and a PC, it would mean the attack is reduced to a 1 x 3 square, instead of a 3 x 3 square. That certainly seems more complicated than just running the rules as written. It might make things easier for the PCs, sure, and you are certainly free to add in house rules to that effect in a game you are running - but it seems a needless change to the rules that only really produces more questions and complications.