ok... for those of you not in the arts ~ the best thing to compliment green: RED.
It will make your eyes pop, and if you have a pale complexion it will work even better. Plus red it the most powerful colour, it makes for the most powerful advertisement (combinations of black red and white). But Brown hair and green eyes, red will look beautiful on you. If you want to go with something less bold, go for another colour bathing suit but have hints of green in it, it will play up your eyes without overwhelming them (if you wore an all green bathing suit it may be too much and your eyes would look dull).
Purple contrasts with green and if it were eye makeup, it would highlight your green eyes. Not sure that it will work in the form of a swimsuit, but you could try some on and see what you think. You also need to be careful of your skintone. What might make your eyes look good, could wash out your skin.