What can I do for my under eye half moons? (Photo)

The treatment that is best for you will need to be decided upon by you and your surgeon.  It is difficult to tell the degree of fullness under the eyes due to the angle of the photo.  You may find benefit from lasers, or fillers.  The more aggressive approach would be surgery.  I encourage you to meet with a board certified plastic surgeon in your area to learn more about your options.

The issue here (based on limited photos) looks to be a contour problem.  Try looking at yourself with overhead lighting while looking straight ahead and then while looking upward.  If you look up and allow the light to get into the hollow areas, you will eliminate the shadows and the appearance of the bags.  How do we produce the same effect?   By increasing the fullness of the hollows with filler injections!

These can be fixed with a combination treatment of: fillers, lasers, and microneedling/PRP can improve under the eyes. We use PRP and stem cells there often. I recommend seeing an expert for a formal consultation. Best, Dr. Emer.

There are several topical and procedural options. To chose between them, you must first be evaluated so the correct treatment for the underlying cause can be utilized. 

There are many options for these concerns, and the "best" is dependent upon many factors that can't unfortunately be answered for your specifically without an in-person consultation.  Non-invasive would include trying some "eye serums" to tighten the skin further over the baggy area, which can help smooth it and lessen the trough under the bagginess (AOX serum by SkinCeuticals is good).    Minimally invasive options would include lower eyelid filler to the trough area to smooth and improve the convexity contour with your upper cheek.  Fraxel treatments can be helpful, but I would go with a coated needle radio-frequency based type to avoid pigmentation changes/risks.  Surgically patients can have some fat removed or transposed (moved) from an unsightly area to a the area that is sunken (the trough).  These are general approaches for lower eyelid concerns.  Getting a qualified consult with a  specialist with high reputation is most suitable in helping making an informed decision.  Best regards.