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

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.

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.

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.

You haven't given much information about yourself, and your images are very limited, so I will simply make a few assumptions here.  Also, it's impossible to provide complete information without a personal examination, so I'll keep my comments mainly general.

First, you seem to be a relatively young person from your images.  Your skin tone is good, and you don't have a  lot of laxity, wrinkles, or hanging tissue.  The contours around your orbits are caused mainly by some hollowing around the bony rim and the fat around the eyeball above it.  This is probably genetic and not age related.  In someone like this, the strategies should be more minimally invasive and directed at manipulation of volume.

My suggestion for this issue is usually to fill in the hollow area with a filler, as this is very reliable, safe, effective, and economical.  If this works and you like it, it's just a matter of keeping up with the fillers about once a year.  If at some point you decide that you would like something more permanent, then transfer of some of your own fat from somewhere else on your body to that area via injections would be a great option.  This will ultimately be a permanent graft, as the fat cells would be living tissue that would remain indefinitely.  The one caveat with this is that you must go to someone who really knows what they are  doing and has a lot of experience with that treatment for optimal results.

A couple of other things worth mentioning are the surgical options for this and the issue of dark circles and pigment.  With regard to the surgical options, you may be a candidate for a procedure called "transconjunctival blepharoplasty," wherein we remove a small amount of bulging fat from the lower eyelid compartment via an incision inside the eye itself in the lower conjunctiva.  This would require a detailed examination to determine though.  While it is a very effective operation in the right person, the risk that we run by doing it is that we make hollowing WORSE by removing fat volume that really needs to be there, just rearranged instead of removed.  If you consider this option, again, be sure you consult with only experienced, properly certified/qualified surgeons who can advise you appropriately.  Regarding the issue of dark circles and pigmentation under the eyes, while this can certainly be accentuated by shadowing resulting from hollows and contour issues, a large degree of this is also simply genetic and involves actual dark pigment either in the skin itself or showing through the very thin and translucent skin from the underlying muscle.  In either event, this will not be addressed by any injections or surgery discussed above, and unfortunately, it is very hard to deal with and sometimes there is no good solution.  I tell you this so you won't have unrealistic expectations of any treatments you might undergo to address the volume alone.  Again, it is important to undergo evaluation and treatment of this area by appropriately qualified doctors to ensure that you are advised correctly about your options for management.  Best of luck.