As mentioned, we have to know the type, colour, extent and location of the birthmark to advise on the most suitable treatment to remove it, be it by laser or surgical methods.
Take for instance, this hyperpigmented blue-black birthmark, called Ota of Naevus, is commonly found on one side on the face around the eye. Pigmentation lasers (eg Q-switched Nd Yag) can be used to treat this birthmark. Several laser sessions are needed and each time, there will be some expected downtime eg redness for a few days.
Another type of birthmark that is red-purplish in colour called Port Wine Stain, can also be found on one side of the face. In this case, vascular lasers are required with several sessions to lighten the birthmark.
So I would recommend you to speak to an experienced doctor in this area to discuss on the most suitable option for you.
Hope this Helps!
Cheers!
Dr David Ng C H