This has actually been a theory of mine for a while now! As always, we may never know for certain. But Alexandria Genesis was not discovered until around the year 2013, [x] thus of course, the people of the colonial days would not know of any proper terminology for this rare sort of diagnosis; but there are many accounts of Hamilton that would actually describe the symptoms. Fisher Ames account of Hamilton's eye color in the New York Mirror was recorded as;
The same writer remarks that General Hamilton’s eyes were grey. This is a mistake. They were of a deep azure, eminently beautiful, without the slightest trace of hardness or severity, and beamed with higher expressions of intelligence and discernment than any others I saw oscillate in the “human face divine.”
Even though Hamilton's own grandson, Allan McLane, would go on to describe Hamilton's eyes as having taken on a purple shade—His grandfather had died years before his birth, so it is far from a first hand account, yet decently worth noting in case he may have heard it from relatives or friends of Hamilton;
His eyes were a deep blue—almost violet—and he undoubtedly presented the physical appearance of his Scotch father rather than his French mother: His eyes were deep set, his nose long, and of the Roman type, and he had a good chin, the jaw being strong; the mouth firm and moderately large.
To go further, other symptoms of Alexandria Genesis were noted in the link above as; “Those who have this mutation will never grow any facial, body, pubic, or anal hair (not including hair on their head, on their ears, noses, eyebrows and eyelashes). Women also do not menstruate, but are fertile”
There are no accounts of Hamilton having any noticable facial hair at all, rather the opposite in fact, where he is often described as having a young, or even feminine appearance. Additionally, if you look closely at many portraits of Hamilton throughout his lifetime, you will notice there are no signs of any facial hair around his portrayed face;
In order; portrait of Alexander Hamilton, by Charles Peale (ca. 1780), Alexander Hamilton portrait, by John Trumbull (ca. 1792), portrait of Alexander Hamilton, by Ezra Ames (ca. 1802)