No, you need to be very gentle when you're cleaning amethyst, and only use very mild soap, water, and a toothbrush. Rinse your amethyst in warm water and dry it with a paper towel. After that, you can use a can of compressed air, like the kind used to clean a computer keyboard, to remove any dirt that's left. However, taking it to a jeweler for a professional cleaning is easier, and they usually don't charge anything.