The Jehovah's witness' version of the bible the NWT translates these verses as:
58 Jesus said to them: “Most truly I say to you, before Abraham came into existence, I have been.” 59 So they picked up stones to throw at him, but Jesus hid and went out of the temple. (emphasis added)
KJV,ESV,NIV,NRSV,NASB all translate the bold section in v58 as "I am", (source) which on the surface, seems to be a direct reference to the way God revealed himself to Moses at the burning bush:
13 Moses said to God, “Suppose I go to the Israelites and say to them, ‘The God of your fathers has sent me to you,’ and they ask me, ‘What is his name?’ Then what shall I tell them?” 14 God said to Moses, “I am who I am. This is what you are to say to the Israelites: ‘I am has sent me to you.’” - Ex 3:14-15 NIV (emphasis added, cf. v14 in different versions)
As such, it would seem logical for the Jews to want to stone Jesus for (in their eyes) blasphemy.
This Q&A on the biblical hermeneutics site addresses which translation of those particular words is preferable, so I don't wish to re-hash that ground, but to ask, given the sense imparted by the following verse (i.e. that the Jews regarded what Jesus was saying as blasphemous) how can it be logically maintained that Jesus was not referencing the name (or title, or description - that particular distinction should be irrelevant) of God as revealed to Moses at the burning bush and applying it to himself?