Power increase at altitude
Well you can set this up to increase the power at altitude - to some extent.
Turbocharged engines have a waste gate, which is a spring-loaded safety valve which prevents overboost - too much boost and you explode the engine. By increasing the spring tension, you increase the allowable pressure in the induction system, but again, it is *very* easy to have too much of a good thing.
This doesn't directly answer the question of why more HP with altitude, but it can - if the waste gate is manually adjustable you can set your HP output to maximum on the ground, and as you climb, increase it somewhat. (You better have a good reason to be in this much of a hurry - mistakes get really expensive, very quickly.)
Turbocharged engines have a "critical altitude" - manifold pressure (and thus HP output) is maintained until the aircraft reaches this critical altitude, which is where the air density is too thin for even the turbocharger to maintain sea level manifold pressure. These can be set up to maintain slightly more (or sometimes considerably more) than sea level manifold pressure, and in cars, often are. We are somewhat more conservative in aircraft, though.
Superchargers, which are direct driven, sometimes have variable gear ratios, known as high blower and low blower - this keeps from overboosting at low altitudes, and allows a change of gear ratio for maximum boost at higher altitudes - there is still a critical altitude, however, even with geared superchargers.
Incidentally, the altitude record for piston engine aircraft is held by NASA - the aircraft is a drone (darn it) powered by a Rotax 912 with FOUR cascaded turbochargers feeding it!
Best Regards,
Echo Mike