What might have influenced Shakespeare to write such an imaginative play as The Tempest?

The Tempest, although listed first in many collections, was the last play written entirely by Shakespeare.  Subsequently,  he contributed to The Two Noble Kinsmen and Henry VIII by John Fletcher. The Tempest was also the only play Shakespeare wrote that didn't have an original source; he himself created the whole plot.  However, the play reflects the news of the day regarding the settlement of the new colony of Virginia, to which England had sent 9 ships to supply the fledgling colony.  The flagship, carrying the new governor of the colony, was separated from the rest during a storm off Bermuda and was given up for lost.  After surviving the shipwreck, the survivors managed to rebuild a boat and sail to 600 mile distant Virginia, arriving nearly a year late.  It was as if they had come back from the dead. The play also appears to have been part of the festive preparations for the marriage of Elizabeth, daughter of James I and Frederick IV of the Palatinate, or around what is now Germany.  The couple were married in 1613.  The Tempest, then, which includes a wedding masque in the middle of the play, was written by commission as a wedding celebration.

Asimov's Guide to Shakespeare, Gramercy Publishing, 2003 ed., pg I-667.