Do I need a separate eye cream?

Not necessarily. A gentle basic facial moisturizer may be enough if it is suitable for the eye area and comfortable on your skin. I would spend based on the concern and formula, not on the idea that every eye area automatically requires a separate category.

Eye cream is optional if a suitable moisturizer already works

Keep it out of your eyes and follow any directions to avoid that area. A separate eye cream can be a preference, but a higher price does not make it an essential step.