Can there be any kind of alternative medicine that can make a black eye go away fast?

No: You can cover it with makeup , but there is no way to get the cells of your body to remove the blood products any faster.

Possbly worth a try: You're dealing with your body's own healing process here, but the homeopathic remedy arnica montana may help. It's very specific for trauma, swelling, and bruising. Though it won't immediately dissolve the bruise, it may stimulate your own restorative mechanisms to go to work. Health food stores often carry 30c potencies; you could safely try a dose every few hours and see how you fare.

Bruising: The best treatment for bruising anywhere, including the eye, is arnica tablests 4 under your tongue every 4 hours while you are awake. It works miracles.