If you dye your hair 2 different colors at once won't the colors bleed together when you wash it?

There is always , in my experience , excess dye that will come off. This usually doesn't matter but in your case you've got the 2 colors. When you wet the hair they will run but will immediately be diluted by the water in the shower . Even if some blue does go on the pink it wont be there long enough to Set into the hair shaft