Can you fly somehow?

Solution 1:

Flying is a cheat/shortcut built into New Game+ (a mode after finishing the game once).

It's available once you get perspective controls back. Press ↑↑↑↑ (up 4x) and jump.

There is no in-game explanation—it's just something you find.

A helpful tip: When flying downwards and triggering Gomez's "falling to his death" animation, you can avoid collapsing upon touching land by jumping just as you make contact.

Solution 2:

I found it by accident as well. Was standing on one of those U shaped pillars (still don't know what they do) and noticed that they looked like a U from all four perspectives. So I hit 'up' four times. Nothing happened, but then I went to jump off the pillar and started flying all over the place!