What is the significance of Jesus saying "I am Alpha and Omega, the beginning and the ending"? All Scriptures are quoted from the King James translation.
Four times in the Revelation Jesus uses that phrase(they are all in red letters):

Revelation 1:8  I am Alpha and Omega, the beginning and the ending, saith the Lord, which is, and which was, and which is to come, the Almighty.
Revelation 1:11  Saying, I am Alpha and Omega, the first and the last: and, What thou seest, write in a book, and send it unto the seven churches which are in Asia; unto Ephesus, and unto Smyrna, and unto Pergamos, and unto Thyatira, and unto Sardis, and unto Philadelphia, and unto Laodicea.
Revelation 21:6  And he said unto me, It is done. I am Alpha and Omega, the beginning and the end. I will give unto him that is athirst of the fountain of the water of life freely.
Revelation 22:13  I am Alpha and Omega, the beginning and the end, the first and the last.

But we know that Jesus was existent at Creation or before since:

Luke 10:18  And he said unto them, I beheld Satan as lightning fall from heaven.

It appears that Jesus had some other motive for using those particular words, rather than Eternal.
This is a powerful assertion of Deity by Jesus.  You are correct that it is, in fact, Jesus Himself who is speaking here and not the Father.  That is made clear in this passage, as Jesus was the One who "was dead" and is now "alive forevermore".


  17 When I saw Him, I fell at His feet like a dead man. And He placed His right hand on me, saying, “Do not be afraid; I am the first and the last, 18 and the living One; and I was dead, and behold, I am alive forevermore, and I have the keys of death and of Hades.  Revelation 1:17-18 NASB 


Jesus is, indeed, the Alpha and the Omega, the First and the Last, the Beginning and the End, the Almighty.

Paul's letter to the Colossians is also simliar:


  He is also head of the body, the church; and He is the beginning, the firstborn from the dead, so that He Himself will come to have first place in everything.   Colossians 1:15 NASB


It is noteworthy to reference the passage in Isaiah where God states something very similar:


  “Who has performed and accomplished it, Calling forth the generations from the beginning?
  ‘I, the Lord, am the first, and with the last. I am He.’”  Isaiah 41:4 NASB


So, it was God (the LORD) in the Old Testament who claimed to be the First and the Last, and it is Jesus Himself who makes that very same claim in the New Testament.  God alone is the First and the Last, and Jesus claims to be God Almighty in an unmistakable manner.