Pope Benedict XVI addressed this in a General Audience on 15 February 2012:
The second word spoken by Jesus on the Cross recorded by St Luke is a word of hope, it is his answer to the prayer of one of the two men crucified with him. The good thief comes to his senses before Jesus and repents, he realizes he is facing the Son of God who makes the very Face of God visible, and begs him; “Jesus, remember me when you come in your kingly power” (v. 42). The Lord’s answer to this prayer goes far beyond the request: in fact he says: “Truly, I say to you, today you will be with me in Paradise” (v. 43). Jesus knows that he is entering into direct communion with the Father and reopening to man the way to God’s paradise. Thus, with this response, he gives the firm hope that God’s goodness can also touch us, even at the very last moment of life, and that sincere prayer, even after a wrong life, encounters the open arms of the good Father who awaits the return of his son.
Dear brothers and sisters, the words of Jesus on the Cross at the last moments of his earthly life offer us demanding instructions for our prayers, but they also open us to serene trust and firm hope. Jesus, who asks the Father to forgive those who are crucifying him, invites us to take the difficult step of also praying for those who wrong us, who have injured us, ever able to forgive, so that God’s light may illuminate their hearts; and he invites us to live in our prayers the same attitude of mercy and love with which God treats us; “forgive us our trespasses and forgive those who trespass against us”, we say every day in the Lord’s prayer.
At the same time, Jesus, who at the supreme moment of death entrusts himself totally to the hands of God the Father, communicates to us the certainty that, however harsh the trial, however difficult the problems, however acute the suffering may be, we shall never fall from God’s hands, those hands that created us, that sustain us and that accompany us on our way through life, because they are guided by an infinite and faithful love.
Note that there is no mention of Today. Eternity is outside time: while the thief was indeed to die "today", he then entered eternity, which cannot be measured in earthly time.
It is entirely possible that the repentant thief went through Purgatory, but any time spent there cannot be measured.
However, the Catechism has more to say:
1030 All who die in God's grace and friendship, but still imperfectly purified, are indeed assured of their eternal salvation; but after death they undergo purification, so as to achieve the holiness necessary to enter the joy of heaven.
Given that Christ himself imparted his grace and friendship to the penitent thief, who are we to say that he was not perfectly purified by those words? It is entirely possible that he did not experience Purgatory at all, and that Christ's today did actually mean today.
One difficulty with timings in earthly time is that Christ descended to the dead before his Resurrection, and then spent six weeks on earth before ascending to his Father. This gives weight to his Today pointing to an Eternity outside earthly time: from an earthly point of view, it certainly wasn't today.
The end result? We don't know. We can be certain of Benedict's interpretation that we have "the firm hope that God’s goodness can also touch us, even at the very last moment of life, and that sincere prayer, even after a wrong life, encounters the open arms of the good Father who awaits the return of his son."