Ask yourself what you want. I find the QIII light is very white, which I love. Many here have modified theirs and have it give off 100 lumens. So, if modding is your thing, the QIII is great for that.
The P1 does have that blue tint, but WOW. So much light for the size. This is one time I have to say smaller is better. I run mine with an r-cr123 that tops off at 4.2v. The P1 is rated for 4v so this is safe, and you can expect 10 - 15 more lumens.
It dims as the battery weakens and falls into normal range. Then the regulator should take over to keep it at 55 lumens.