The scholars who have most meticulously studied Hamilton's religion conclude that though he probably had a conventional religious youth, during the time in which he acted as a "Founder" (from 1776-after 1800) he was not conventionally religious. In other words, he was not an orthodox Trinitarian Christian. (For more see here.)
After his son Philip died in a duel in 1801, Hamilton, understandably grief stricken, converted to a less generic/more concrete religion -- orthodox Christianity.
What follows are some primary sources on Hamilton's death. Interestingly though he had accepted the truth of the orthodox Christian religion by his death in 1804, he had never gotten around to joining a church, which suggests the "newness" of Hamilton's orthodox faith.
Also of note, Hamilton converted to a form of orthodox Christianity that, understandably, held communion to be a central sacrament. Yet, when he begged for the Lord's Supper on his deathbed, Hamilton appeared naively unaware of the rituals surrounding administering the sacrament in the two churches from which he sought communion: The Episcopalian and the Presbyterian. As we will see, Hamilton was TWICE refused communion on his deathbed before, finally, having it administered.
On his deathbed Hamilton first asked Bishop Benjamin Moore of the Protestant Episcopal Church and was initially refused. As Moore recounts:
Yesterday morning, immediately after he was brought from Hoboken to the house of Mr. B yard, at Grenwich, a message was sent informing me of the sad event, accompanied by a request from General Hamilton, that I would come to him for the purpose of administering the holy communion. I went, but being desirous to afford time for serious reflection, and conceiving that under existing circumstances, it would be right and proper to avoid every appearance of precipitancy in performing one of the most solemn offices of our religion, I did not then comply with his desire. ...
Hamilton then sent for the Presbyterian Dr. John Mason and was again refused. Here is how Mason recounts the incident:
... The exchange of melancholy salutation, on entering the General’s apartment, was succeeded by a silence which he broke by saying, that he “had been anxious to see me, and have the sacrament administered to him; and that this was still his wish.” I replied, that “it gave me unutterable pain to receive from him any request to which I could not accede: that, in the present instance, a compliance was incompatible with all my obligations; as it is a principle in our churches never to administer the Lord’s supper privately to any person under any circumstances.” He urged me no further. I then remarked to him, that “the holy communion is an exhibition and pledge of the mercies which the Son of God has purchased; that the absence of the sign does not exclude from the mercies signified; which were accessible to him by faith in their gracious Author.” “I am aware,” said he, “of that. It is only as a sign that I wanted it.”...
Bishop Moore was then called back and finally administered the Lord's Supper. As he noted:
... At one o'clock I was again called on to visit him. Upon my entering the room and approaching his bed, with the utmost calmness and composure he said, "My dear sir, you perceive my unfortunate situation, and no doubt have been made acquainted with the circumstances which led to it. It is my desire to receive communion at your hands. I hope you will not conceive there is any impropriety in my request."
He added, "It has for some time past been the wish of my heart, and it was my intention to take an early opportunity of uniting myself to the church, by the reception of that holy ordinance." I observed to him, that he must be very sensible of the delicate and trying situation in which I was then placed: that however desirous I might be to afford consolation to a fellow mortal in distress; still, it was my duty, as a minister of the Gospel, to hold up the law of God as paramount to all other law: and that, therefore, under the influence of such sentiments, I must unequivocally condemn which had brought him to his present unhappy condition. He acknowledged the propriety of these sentiments, and declared that he viewed the late transaction with sorrow and contrition. I then asked him, "Should it please God to restore you to health, sir, will you never be again engaged in a similar transaction? And will you employ all your influence in society to discountenance this barbarous custom?" His answer was "That, sir, is my deliberate intention."
I proceeded to converse with him on the subject of receiving the communion; and told him that with respect to the qualifications of those who wished to become partakers of that holy ordinance, my enquiries could not be made in language more expressive than that which was used by our church. "Do you sincerely repent of your sins past? Have you a lively faith in God's mercy thro' Christ, with a thankful remembrance of the death of Christ? And are you disposed to live in love and charity with all men?" He lifted up his hands and said, "With the utmost sincerity of heart I can answer those questions in the affirmative. I have no ill will against Col. Burr. I met him with a fixed resolution to do him no harm. - I forgive all that happened." I then observed to him, that the terrors of the divine law were to be announced to the obdurate and impenitent, but that the consolations of the Gospel were to be offered to the humble and contrite heart; that I had no reason to doubt his sincerity and would proceed immediately to gratify his wishes. The Communion was then administered, which he received with great devotion, and his heart afterwards appeared to be perfectly at rest. I saw him again this morning, when with his last faltering words he expressed a strong confidence in the mercy of God through the intersession of the Redeemer. I remained with him until two o'clock this afternoon, when death closed the awful scene. - he expired without a struggle, and almost without a groan.