P.W.
Anything permanent, like a backsplash, keep pretty neutral. You don't have to go beige, but one day you will be sick of purple, and then you will be stuck with a purple backsplash.
Put hints of color in your backsplash, so that you can repaint your walls when you are sick of them (and it will happen), and the backsplash won't dictate what color you can paint. I had bright purple and yellow in a living room once, and I certainly wouldn't want that now.
Pick something with subtle multi-color, to give you options.