if you have waterproof makeup chances are you do need an eye makeup remover to get that makeup off your eyes. if you are using water-based non-waterproof then you can use your general cleanser to get your makeup off. when you get that raccoon eyes you do need to wash afterwards because that is makeup left on your skin so you do still need to wash afterwards to get all that makeup removed from around your eyes.
my makeup remover gives me raccoon eyes when i remove mascara. can't i just wash my makeup off with soap and water?