This paper investigates — in the light of his paper On quaternions and the rotation of a solid body, dating from 1848 — several algebraic concepts (endomorphism, conjugation, characteristic polynomial) that played a major role in the ultimate phase of Hamilton's work. Looking into the case of duality, it will be seen that this provides an opportunity of making apparent how his mathematical practices were linked to his readings and optical or physical investigations, harking back to the 1830s.