Since Prospero orchestrates events, saying his luck has finally taken a turn for the better, an analogy between Prospero and playwright can certainly be drawn: a playwright orchestrates all events in a theatrical work. Such an analogy draws more strength from the romance that Prospero manipulates between Miranda and Ferdinand as it parallels the romances that Shakespeare is so famous for, such as that between Rosalind and Orlando.
It's important to understand why the play was written. In 1613, Princess Elizabeth (named in honor of Elizabeth I) was married to Prince Frederick of the Palatinate (later Germany) at the Palace of Whitehall. There's good evidence that the Tempest was written as a celebratory event for the wedding, and that Shakespeare did indeed create characters in the play from the guests in attendance, including himself as Prospero.
The Tempest was not Shakespeare's last play; he collaborated with John Fletcher (1579-1625) on Henry VIII and The Two Noble Kinsmen.
Asimov's Guide to Shakespeare, Gramercy Publishing, (c) 1970, pg. I-670.
Prospero has long been regarded as an alter ego of Shakespeare, largely because of the famous speech in which Propsero proclaims,
Our revels now are ended. These our actors,
As I foretold you, were all spirits and
Are melted into air, into thin air:
And, like the baseless fabric of this vision,
The cloud-capp'd towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Ye all which it inherit, shall dissolve
And, like this insubstantial pageant faded,
Leave not a rack behind.
The reference to the "great globe" has often been read as an allusion to the Globe Theatre, and The Tempest has often been read as a kind of valediction by Shakespeare to his theatrical career.
It should be noted, however, that some critics regard this view as an over-simplification, and it should also be noted that some critics regard Prospero as a highly unattractive figure.
I definitely think that it is possible to regard Prospero as Shakespeare in disguise. Let us remember that on the island where the play is set, Prospero, through his magic, acts as a kind of director or stage manager, where he effectively controls all the characters and ensures that they experience what he wants them to. It is Prospero who has the control of everything, and he chooses to place characters in certain situations. Let us also remember that #2 is correct in stating that this play was thought to be Shakespeare's last. It is therefore highly symbolic that Prospero gives up his magic as he anticipates returning to "normal" life. There is obviously a parallel between this and Shakspeare renouncing his "magic" as a playwright.
If Prospero is not supposed to represent Shakespeare himself, then the character certainly suggests a representation of Shakespeare the playwright and poet. Since The Tempest was most likely The Bard's final play, he seems to be bidding a farewell to the stage through Prospero's comments in the final act.
I'll break my staff,
Bury it certain fathoms in the earth,
And deeper than did ever plummet sound
I'll drown my book.
(Act V, Scene i, 54-57)