Eye creams/serums are a marketing gimmick and a sure sign that a company does not have your best interest in mind!
The skin around your eyes is very different from the rest of the skin on your face. It's much more reactive and delicate and...it's next to your eyes. Thus, the majority of eye-focused products on the market contain ingredients that don't trigger this reaction but also don't do much else. Let's take collagen peptides and caffeine––two of the most common eye-cream ingredients––for instance. Putting collagen ON your skin is like putting gasoline ON your car. Your skin can't absorb or utilize it other than as a basic moisturizer. Collagen must be produced from within by things like Vitamin C and retinoids! Caffeine, another common ingredient, is included because of this logic of helping with "tired" looking eyes. Topical caffeine functions as an antioxidant but one that is only a fraction as effective as Vitamin C. Thus, it's best to stick with basic moisture around your eyes or to (ask your doctor and) apply the same go-to active ingredients that you use on the rest of your skin, like Vitamin C and a good retinoid!