While we are looking at possible ways to understand ἐγώ εἰμι, it's worth pointing to these two uses earlier in the same chapter:
John 8:24 εἶπον οὖν ὑμῖν ὅτι ἀποθανεῖσθε ἐν ταῖς ἁμαρτίαις ὑμῶν· ἐὰν γὰρ μὴ πιστεύσητε ὅτι ἐγώ εἰμι, ἀποθανεῖσθε ἐν ταῖς ἁμαρτίαις ὑμῶν.
John 8:28 εἶπεν οὖν ὁ Ἰησοῦς· Ὅταν ὑψώσητε τὸν υἱὸν τοῦ ἀνθρώπου, τότε γνώσεσθε ὅτι ἐγώ εἰμι, καὶ ἀπ’ ἐμαυτοῦ ποιῶ οὐδέν, ἀλλὰ καθὼς ἐδίδαξέν με ὁ πατὴρ ταῦτα λαλῶ.
Especially since both are part of a passage that leads up to the one we are talking about here:
21 Εἶπεν οὖν πάλιν αὐτοῖς· Ἐγὼ ὑπάγω καὶ ζητήσετέ με, καὶ ἐν τῇ ἁμαρτίᾳ ὑμῶν ἀποθανεῖσθε· ὅπου ἐγὼ ὑπάγω ὑμεῖς οὐ δύνασθε ἐλθεῖν. 22 ἔλεγον οὖν οἱ Ἰουδαῖοι· Μήτι ἀποκτενεῖ ἑαυτὸν ὅτι λέγει· Ὅπου ἐγὼ ὑπάγω ὑμεῖς οὐ δύνασθε ἐλθεῖν; 23 καὶ ἔλεγεν αὐτοῖς· Ὑμεῖς ἐκ τῶν κάτω ἐστέ, ἐγὼ ἐκ τῶν ἄνω εἰμί· ὑμεῖς ἐκ τούτου τοῦ κόσμου ἐστέ, ἐγὼ οὐκ εἰμὶ ἐκ τοῦ κόσμου τούτου. 24 εἶπον οὖν ὑμῖν ὅτι ἀποθανεῖσθε ἐν ταῖς ἁμαρτίαις ὑμῶν· ἐὰν γὰρ μὴ πιστεύσητε ὅτι ἐγώ εἰμι, ἀποθανεῖσθε ἐν ταῖς ἁμαρτίαις ὑμῶν. 25 ἔλεγον οὖν αὐτῷ· Σὺ τίς εἶ; εἶπεν αὐτοῖς ὁ Ἰησοῦς· Τὴν ἀρχὴν ὅ τι καὶ λαλῶ ὑμῖν; 26 πολλὰ ἔχω περὶ ὑμῶν λαλεῖν καὶ κρίνειν· ἀλλ’ ὁ πέμψας με ἀληθής ἐστιν, κἀγὼ ἃ ἤκουσα παρ’ αὐτοῦ ταῦτα λαλῶ εἰς τὸν κόσμον. 27 οὐκ ἔγνωσαν ὅτι τὸν πατέρα αὐτοῖς ἔλεγεν. 28 εἶπεν οὖν ὁ Ἰησοῦς· Ὅταν ὑψώσητε τὸν υἱὸν τοῦ ἀνθρώπου, τότε γνώσεσθε ὅτι ἐγώ εἰμι, καὶ ἀπ’ ἐμαυτοῦ ποιῶ οὐδέν, ἀλλὰ καθὼς ἐδίδαξέν με ὁ πατὴρ ταῦτα λαλῶ. 29 καὶ ὁ πέμψας με μετ’ ἐμοῦ ἐστιν· οὐκ ἀφῆκέν με μόνον, ὅτι ἐγὼ τὰ ἀρεστὰ αὐτῷ ποιῶ πάντοτε. 30 ταῦτα αὐτοῦ λαλοῦντος πολλοὶ ἐπίστευσαν εἰς αὐτόν.
And for an example that does not refer to Jesus, here's one from the next chapter:
John 9:9 ἄλλοι ἔλεγον ὅτι Οὗτός ἐστιν· ἄλλοι ἔλεγον· Οὐχί, ἀλλὰ ὅμοιος αὐτῷ ἐστιν. ἐκεῖνος ἔλεγεν ὅτι Ἐγώ εἰμι.
How likely is it that ἐγώ εἰμι is used in the same sense in these three verses and in John 8:58?
ἐξίσταντο δὲ πάντες καὶ διηποροῦντο, ἄλλος πρὸς ἄλλον λέγοντες, τί θέλει τοῦτο εἶναι;
http://jonathanrobie.biblicalhumanities.org/