Someone is trying to sell me alkaline stick and claims that it turns tap or bottled water into alkaline water which has health benefits like being an antioxidant, frees radicals, anti-aging, etc. I don’t believe at all.

I suppose it depends on whom you believe! This is what wiki says ::
Some research suggests that alkaline reduced water may be useful in scavenging free radicals in the laboratory setting. Tests on in vitro lymphocytes suggest that reduced water can prevent hydrogen peroxide-induced damage to DNA, RNA and certain proteins. However, drinking ionized water would not be expected to alter the body’s pH, and there is no evidence of any claims made by manufacturers that drinking ionized water will have a noticeable effect on the body.

Electrolyzed water has been used by the food industry to sanitize food products; though effective in bacterial solutions, it was found less useful when sanitizing utensils, surfaces and food products.

http://en.wikipedia.org/wiki/Alkaline_water

Ok so I have dark circles under my eyes, I sleep well, eat well, not overweight, I don’t smoke, I drink occasionally but not often. I think its hereditary.
So my mom told me to use her anti-age eye cream every night which is rich in antioxidants, but it seems to be darkening the skin under my eyes, how can this happen?

don’t know why it would darken the circles under your eyes, but I can tell you from what you wrote that the solution to your problem is to drink more water.