If they're the same person, technically he chased 'himself' with an axe.
As far as whether it's cosmic horror, like
Stinkles brought up, the crux of that genre is whether the threat is both fully knowable and conquerable. Cosmic horror requires one of these to be "no".
This wouldn't qualify because they are in a Grecian purgatory, meaning their situation could be explained to them. And if this situation is knowable, they can conquer it if Young Tommy would stop making the same mistakes like killing the gull. Purgatory is ultimately a test to determine where you end up. (Technically, they could conquer it without knowledge, but it'd be easier if they knew they were being judged.)
The only argument in favor of it being cosmic horror is that the lighthouse beacon's forbidden knowledge fries your brain. It has Lovecraftian elements but it's not cosmic horror.
Contrast this with The Witch, which has no Lovecraftian elements but is cosmic horror, because the threat is knowable but is NOT conquerable. That family is 150% fucked.