In the first chapter of Revelation we read:
“I am the Alpha and the Omega,” says the Lord God, “who is and who was and who is to come, the Almighty.” (Rev 1:8)
Then in the 21st chapter we read:
5 And he who was seated on the throne said, “Behold, I am making all things new.” Also he said, “Write this down, for these words are trustworthy and true.” 6 And he said to me, “It is done! I am the Alpha and the Omega, the beginning and the end. To the thirsty I will give from the spring of the water of life without payment. 7 The one who conquers will have this heritage, and I will be his God and he will be my son. (Revelation 21:5-7)
The above is unambiguously about God.
But then, in the next, 22nd, chapter, it appears to be Jesus speaking when it says “I am the Alpha and the Omega, the first and the last, the beginning and the end.” (Rev 22:13) The full text is as follows:
8 I, John, am the one who heard and saw these things. And when I heard and saw them, I fell down to worship at the feet of the angel who showed them to me, 9 but he said to me, “You must not do that! I am a fellow servant with you and your brothers the prophets, and with those who keep the words of this book. Worship God.”
10 And he said to me, “Do not seal up the words of the prophecy of this book, for the time is near. 11 Let the evildoer still do evil, and the filthy still be filthy, and the righteous still do right, and the holy still be holy.”
12 “Behold, I am coming soon, bringing my recompense with me, to repay each one for what he has done. 13 I am the Alpha and the Omega, the first and the last, the beginning and the end.”
14 Blessed are those who wash their robes,[c] so that they may have the right to the tree of life and that they may enter the city by the gates. 15 Outside are the dogs and sorcerers and the sexually immoral and murderers and idolaters, and everyone who loves and practices falsehood.
16 “I, Jesus, have sent my angel to testify to you about these things for the churches. I am the root and the descendant of David, the bright morning star.”
17 The Spirit and the Bride say, “Come.” And let the one who hears say, “Come.” And let the one who is thirsty come; let the one who desires take the water of life without price. (Revelation 22:8-17)
Since all of this is the angel speaking it is possible that the angel speaks on behalf of God in 12-13 but “Behold I am coming soon” would more naturally introduce words of Jesus and from 16 “I, Jesus, have sent my angel…” makes this explicit.
The point is though that the this section still speaks in terms of delegated authority. It does not contradict Jesus’ repeated statements “my God” referring to his God. So it is more natural to see verse 13 “Alpha and Omega” as simply another instance where Jesus takes on many of the titles of his father. This was explained by Paul in Philippians:
Therefore God has highly exalted him and bestowed on him the name that is above every name, 10 so that at the name of Jesus every knee should bow, in heaven and on earth and under the earth, 11 and every tongue confess that Jesus Christ is Lord, to the glory of God the Father. Philippians 2:9-11
When Jesus took on some of his father’s work, he also took on his titles, and even says Paul God’s name – although in fact Alpha and Omega is a title not a name.
For a related topic see also : What about language applied to God in the Old Testament but applied to Jesus in Revelation?