Believe it or not, there is no magic to the words ἐγὼ εἰμί . In the bible, it always functions as a pronoun-verb of simple identification. Jesus isn't the only one who uses the words ἐγὼ εἰμί. Here are some examples:
καταβὰς δὲ Πέτρος πρὸς τοὺς ἄνδρας εἶπεν Ἰδοὺ ἐγώ εἰμι ὃν ζητεῖτε· τίς ἡ αἰτία δι’ ἣν πάρεστε;
Acts 10:21
ἄλλοι ἔλεγον ὅτι Οὗτός ἐστιν· ἄλλοι ἔλεγον Οὐχί, ἀλλὰ ὅμοιος αὐτῷ ἐστιν. ἐκεῖνος ἔλεγεν ὅτι Ἐγώ εἰμι.
John 9:9
καὶ λυπούμενοι σφόδρα ἤρξαντο λέγειν αὐτῷ εἷς ἕκαστος Μήτι ἐγώ εἰμι, Κύριε;
Matthew 26:22
καὶ γὰρ ἐγὼ ἄνθρωπός εἰμι ὑπὸ ἐξουσίαν, ἔχων ὑπ’ ἐμαυτὸν στρατιώτας, καὶ λέγω τούτῳ Πορεύθητι, καὶ πορεύεται, καὶ ἄλλῳ Ἔρχου, καὶ ἔρχεται, καὶ τῷ δούλῳ μου Ποίησον τοῦτο, καὶ ποιεῖ.
Matthew 8:9
καὶ γὰρ ἐγὼ ἄνθρωπός εἰμι ὑπὸ ἐξουσίαν τασσόμενος, ἔχων ὑπ’ ἐμαυτὸν στρατιώτας, καὶ λέγω τούτῳ Πορεύθητι, καὶ πορεύεται, καὶ ἄλλῳ Ἔρχου, καὶ ἔρχεται, καὶ τῷ δούλῳ μου Ποίησον τοῦτο, καὶ ποιεῖ.
Luke 7:8
καὶ εἶπεν Ζαχαρίας πρὸς τὸν ἄγγελον Κατὰ τί γνώσομαι τοῦτο; ἐγὼ γάρ εἰμι πρεσβύτης καὶ ἡ γυνή μου προβεβηκυῖα ἐν ταῖς ἡμέραις αὐτῆς.
Luke 1:18
καὶ ἀποκριθεὶς ὁ ἄγγελος εἶπεν αὐτῷ Ἐγώ εἰμι Γαβριὴλ ὁ παρεστηκὼς ἐνώπιον τοῦ Θεοῦ, καὶ ἀπεστάλην λαλῆσαι πρὸς σὲ καὶ εὐαγγελίσασθαί σοι ταῦτα·
Luke 1:19
ὡς δὲ ἐπλήρου Ἰωάνης τὸν δρόμον, ἔλεγεν Τί ἐμὲ ὑπονοεῖτε εἶναι, οὐκ εἰμὶ ἐγώ· ἀλλ’ ἰδοὺ ἔρχεται μετ’ ἐμὲ οὗ οὐκ εἰμὶ ἄξιος τὸ ὑπόδημα τῶν ποδῶν λῦσαι.
Acts 13:25
Here are some more examples -- Acts 10:26, Acts 22:3, Acts 23:6, Acts 26:29, Romans 7:14, Romans 11:1, Romans 11:13, 1 Cor. 15:9 , 1 Timothy 1:15, 1 Cor. 1:12 , 1 Cor. 3:4, John 3:28 , Acts 13:25 , John 18:35, Matthew 26:25
------
Here's Kenneth L. McKay :
Kenneth L. McKay, who graduated with honors in Classics from the
Universities of Sydney and Cambridge, taught Greek in universities
and theological colleges in Nigeria, New Zealand, and England, who
taught at the Australian National University for 26 years, has written
numerous articles on ancient Greek syntax, as well as authored a
book on Classical Attic, Greek Grammar for Students, and A New
Syntax of the Verb in New Testament Greek: an aspectual approach,
provides the following in relation to the alleged "true parallel
between Exodus 3:14 (LXX) and John 8:58":
'I am' in John's Gospel
BY K. L. MCKAY,
It has become fashionable among some preachers and writers to
relate Jesus's use of the words 'I am' in the Gospel according to John,
in all, or most, of their contexts, to God's declaration to MOSES in
Exodus 3:14, and to expound the passages concerned as if the words themselves
have some kind of magic in them. Some who have no more than a smattering of
Greek attribute the 'magic' to the Greek words ego eimi. (1) I wish briefly
to draw attention to the normality of the Greek in all such passages, and the
unlikelihood of the words ego eimi being intended to suggest any special
significance of this kind.
<...>
The verb `to be' is used differently, in what is presumably
its basic meaning of `be in existence,' in John 8:58: _prin
Abraam genesthai ego eimi, which would be most naturally
translated `I have been in existence since before Abraham
was born,' IF IT WERE NOT FOR THE *OBSESSION* WITH THE
SIMPLE WORDS `I AM.' If we take the Greek words in their
natural meaning, as we surely should, the claim to have been
in existence for so long is in itself a staggering one,
quite enough to provoke the crowd's violent reaction
. . . So the emphatic words used by Jesus in the passages
referred to above [one of which is John 8:58] are perfectly
natural in their contexts, AND THEY DO NOT ECHO THE WORDS OF
EXODUS 3:14 IN THE NORMALLY QUOTED GREEK VERSION. Thus they
are quite UNLIKELY to have been used in the New Testament to
convey that significance, however much the modern English
versions of the relevant passages, following the form of the
Hebrew words, may suggest it. (K. L. McKay, "`I am' in
John's Gospel," Expository Times (1996): 302-303)
(1) I have seen one such speaker try to impress his audience by writing the
words on a blackboard, only to demonstrate that he was ignorant of even the
simplest details of Greek.
Οὐαὶ οἱ λέγοντες τὸ πονηρὸν καλὸν καὶ τὸ καλὸν πονηρόν, οἱ τιθέντες τὸ σκότος φῶς καὶ τὸ φῶς σκότος, οἱ τιθέντες τὸ πικρὸν γλυκὺ καὶ τὸ γλυκὺ πικρόν