Amethyst can be cleansed in a variety of ways. Smudging with sage, lavender or palo santo is a great way to cleanse your amethyst. Using a tuning fork will also do the trick. A tap or two will knock off any yuckie energy. Also, since Amethyst has Mohs Hardness of 7.0, you can cleanse this stone with salt water as well. Soak your crystal in bowl of water with sea salt for at least 20 to 30 minutes.
You can leave it in the salt water for as long as you feel comfortable, some people leave it for as many as 3 days. Trust your intuition on the time you spend soaking. Finally, leaving your Amethyst under a Full Moon on your windowsill will cleanse it fully and thoroughly. No matter what way you chose to cleanse your amethyst, just be sure to set an intention that your crystal be cleansed with all that does not belong to you.