Sam- not commenting on conclusions here, but yes it was the custom of many in that era to be baptized upon death bed in the belief that baptism remitted sin, and it increased the likelihood of dying without sin. And it is a common belief Constantine did just that.
http://www.answers.com/topic/constantine-i-the-great
Above link is actually a compilation of several articles on Constantine but pertinent to your observation:
In 326, Constantine had his eldest son Crispus tried and executed, as he believed accusations that Crispus had been having an affair with Fausta, Constantine's second wife. A few months later he also had Fausta killed as the apparent source of these false accusations.
Eusebius reports that Constantine was baptized only shortly before his death in 337. He moved from the Capital to a neighboring thermal spa to take the waters, and thence to his mother's city of Helenopolis, where he prayed in the great church that she built in honor of Lucian the apostle. With this, he followed one custom at the time which postponed baptism until old age or death[19]
19. In this period infant baptism, though practiced (usually in circumstances of emergency) had not yet become a matter of routine in the west. See Thomas M. Finn (1992), Early Christian Baptism and the Catechumenate: East and West Syria. See also Philip Rousseau (1999). "Baptism", in Late Antiquity: A Guide to the Post Classical World, ed. Peter Brown.
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