Latisse is a drug from the prostaglandin class of medications. It is the only FDA-approved medication to grow eyelashes. Do other glaucoma medications enhance eyelash growth? Absolutely, all of the prostaglandins do (Travatan & Xalatan), however, keep in mind that the companies that make the other drops have not gone back to the FDA to seek approval for this clinical use, so we cannot prescribe these medicines for eyelash growth. Only those "lucky" patients who use these for glaucoma will see the eyelash-growing benefits. Cosmetic patients will have to stick to Latisse.
There is no difference between Lumigan (the original formula) and Latisse. They are both bimatoprost 0.03%. I mention the original formula, because Allergan has recently released a newer version of Lumigan which is 0.01% to try to reduce eye redness (one of the most common and bothersome side effects for patients who use Lumigan for glaucoma).
Does combigan cause eyelashes to grow? NEVER. Combigan is combination of brimonidine (an alpha-agonist) and timolol (a beta-blocker). Neither of these classes of drugs will cause eyelash growth.
Bottom line: don't self-medicate. Get your medications from a reputable doctor and you will get the right medication for the right indication.