Registering to try and bring some clarity here.
The HD800 are audiophile level headphones. There is no denying that within the HiFi community they are known to be extremely neutral which results in some people calling them "passionless". They are known for having one of the best if not the best(subjective) soundstages out there. They arent without their detractors nearly all of which is subjective and is highly connected to what content the headphones are being used for
Due to the fact that its a standard dynamic driver the headphones they can be a bit lacking in one of the most visceral ranges of audible sound - low end. The HD800 will do a great job letting you hear the a stand-up bass or a double bass but it will absolutely be lacking when it comes to feeling an explosion, especially if your coming from even a rudimentary setup with something that can move a lot of air like a sub woofer or high end tower speakers. The lack of impact is further exacerbated by the fact that HD800 is an open design - This greatly improves the sound stage and clarity but due to the fact that you don't basically have little tiny room on your head containing the sound waves the "boom" of explosions will be lessened.
Amp wise, yes you can run the HD800 off a Magni/Modi stack, yes you can run it off an iPhone. Sound quality will change depending on this. While many headphones can run from many different amps in many instances the amount of power the amp puts out will greatly affect the sound quality. For the coil in a dynamic driver to move the diaphragm you need energy, this can be a little or a lot(to a limit). Just as with a car where there is an optimal fuel/air mixture in the engine there is also an optimal amount of fuel(electricity)/air mixture in the headphone. To react to the impulse of the audio input signal you must have enough energy to transition between different frequencies as fast as needed. Having low amounts of power can result in flat sounding and muddy noise.
So yes - I would consider the Magni/Modi to be a minimal setup for the HD800 but if you keep on reading you'll find out why it doesn't actually matter especially in the context of gaming for the HD800.
Source, source, source. This is the king of all measures of how good something sounds. If your source sounds like crap its going to sound like crap no matter what you listen to it on. In fact in some cases it will sound even worse on high end gear because of the revealing nature of said gear. You'll hear compression, frequency cutoffs, gain issues, ect.
To the OP I would submit this advice.
- Listen to them first at a store
- Listen to them again at a store
- The HD800 will likely have a price drop soon with the HD800S announced recently
- Look into other headphones if gaming is your focus such as the HiFiMAN HE560 which is less expensive and will give you better "booms" in games.
Disclaimer: I am an audiophile. I've spent far more than I care to admit on gear across the range "just to see". My current headphone stack is Schiit Yggdrasil > Cavalli Liquid Carbon > HiFiMAN HE1000.