The first thing that comes to mind when I think "off-beat platformer" is Earthworm Jim. This game isn't even close to being a fine-tuned efficient action experience. But it makes up for that with a ton of heart and creativity. The game often ups the ante with scenarios and stages you'll never see coming.

Even if it isn't the most well-designed platformer, the game is still a lot of fun from beginning to end. Also, in my opinion, I prefer the SNES version. Genesis had an extra stage, but I didn't find it enjoyable at all. I also think SNES' "rougher" sound chip suited the vibe of the game a lot more.