Cool stuff. I accidentally got myself stuck here thanks to the ledge-grabbing upgrade =D
I played this for quite a while and really enjoyed it. There are some little things that I think can be improved a bit. One thing is that the character leaps forward when you jump while moving, even if you just pressed the move button for a bit. This caused me to fall a lot when I am standing on a small platform and there is a monster next to me that I want to kill by jumping on it. So the jumping should be consistent.
The other thing is the double/triple jump upgrade. I didn't find it very useful or I am just bad at utilizing it. I think It might be better if it was a normal extra jump in mid-air, instead of landing and then jumping again a higher distance. But that might require redesigning some areas since it may render the edge-grabbing useless.
Very good game overall and cool concept. Good job!