All other things being equal, immersion brewing, including aeropress, cold brew, French press, hario switch, etc., leads to a drink with more caffeine than other brewing methods.
Aeropress has much more in common with espresso than cold brew or French press, as the immersion step doesn't last very long (based on how it was made for me).
In fact, cold press is pretty much the only outlier here. Caffeine is highly soluble in water, and the low temperature is the only thing slowing the extraction process down in a cold brew. In all of the other examples, the caffeine is mostly (though admittedly not entirely) extracted before most of the flavors are.
