Is it normal to experience extreme eye swelling after a Cosmelan treatment?

Treating PIH and melasma with cosmelan may make your skin hypersensitive. I would suggest Melarase creams and Elevase Moisture Booster as part of the post-treatment regimen to reduce redness and pigmentation risk.
Best, 
Dr. Karamanoukian
Realself100 Surgeon

It is normal for some patients to experience swelling after the cosmelan treatment & around the eye area. It usually is not from an allergic reaction.

Some swelling after Cosmelan treatment is acceptable, but if the swelling is severe allergic reaction must be considered.

NO there should be no swelling if used as directed.
You probably are having an allergic reactio to one of the ingredients. See your internist to treat the allergic reaction.

In addition to a variety of acids, Cosmelan also contains botanicals that can incite allergic reactions. To answer your question, it is NOT normal for you to be experiencing what appears to be an allergic reaction. I would seek the care of your doctor as soon as you are able to do so. If you are coughing or experiencing difficulty breathing, you should go to the emergency room immediately.

This certainly sounds like an allergic reaction.
Cosmelan has a large variety of ingredients including kojic acid, arbutin, azelaic acid, etc. any of which can cause an allergic reaction
Hopefully, this peel was done in a physician's office where the treatment of such a reaction should be quite routine. If so, I would return there.
If your peel was done in a so-called medi-spa, I would recommend a visit to a dermatologist for treatment. Unfortunately, we are becoming more and more experienced in treating untoward incidents perpetrated by our more avaricious colleagues lured into the cosmetic realm.
As Dr. Chen advises, if you are having shortness of breath or weakness, head straight to an ER.

 I have no experience with this product but I can still say for sure that this is not "normal" and your doctor would want to know about what you are experiencing asap.  You might have an allergic reaction and hopefully not an infection.  Call them right away for an evaluation.