Jesus called himself "I am" a direct reference to God's statement to Moses from the burning bush:

And God said to Moses, "I AM WHO I AM." And He said, "Thus you shall say to the children of Israel, 'I AM has sent me to you.'" (Exodus 3:14)

"Jesus said to them, "Most assuredly, I say to you, before Abraham was, I AM." (John 8:58)

It is clear from the context that this is just exactly how Jesus meant his statement. It is also clear that His hearers took this statement as a claim to be God because they immediately picked up stones to stone Him. (Stoning was the penalty under Jewish law for blasphemy against the name of God (Exodus 24:11-16)