Your square doesn't have to (and arguably shouldn't) match your tie perfectly. Any kind of burgundy would work fine, or any other colors in the tie (like the eggplant or charcoal you mentioned). A white linen, or even better a beige/light grey/cream/off-white/whatever silk square to match your shirt would also work. Additionally, if your suit has any checks or stripes, you could match your square to the colors in that.
It really doesn't matter that much, as long as it matches/compliments something else in your outfit. Don't over-think it.