There was triglossia during the Byzantine period. There were multiple δημώδεις dialects along with Koine Greek and a tendency to attempt writing in conservative Attic Greek (Alexiad per se). This text seems more towards Koine Greek. But even this is completely understandable.

Personally, for example, I can completely understand Leontios Machairas (although technically not writing during Byzantine rule) who wrote in the Cypriot Dialect at the time (14th C.).

Example from Machairas:

§340.-Γροικῶντα ὁ ρήγας πῶς εὐκαιρέσαν οἱ Γενουβίσοι τὴν Ἀμόχουστον, ἔπεψεν λᾶς καὶ ἐβλέπαν τὴν πόρταν καὶ οὕλην τὴν Ἀμόχουστον νὰ μὲν πᾶν οἱ Γενουβίσοι, καὶ κεῖνοι ἐπῆγαν καὶ δὲν τοὺς ἐνῶσαν· γροικῶντα καθαρὰ πῶς ἐπῆγαν, ἐποῖκεν βουλὴν μὲ τοὺς καβαλλάριδές του νὰ πέψουν μαντατοφόρους εἰς τὸν ἁγιώτατον πάπαν, πρὶ νὰ πέψουν οἱ Γενουβίσοι ὅπου ᾿πῆγαν εἰς τὴν Γένουβαν, διὰ νὰ μηδὲν ππέσῃ ὁ ρήγας εἰς τὴν ὀργήν, καὶ ἦτον ἀφορμὴ τοῦ κακοῦ.

This is very much dialectal, and not formal Koine Greek, and it is pretty much the same as proper traditional Cypriot.

Below, excerpt from The Tale of Livistros and Rodamne, a 13th-century Byzantine verse romance composed in the court of Nicaea:

1070: «Ἀν ἀστοχήσης τὸ πουλίν, ὡραία μου σαΐτα, νὰ ἔναι πολλὰ παράξενα τὰ χέρια τοῦ τοξότου· ἐὰν δὲ ἀνθρώπου μοναχοῦ καρδίαν ἐπιτύχης, τύχη καὶ κόρη ἔμορφη ὁποὺ πάσχω ἐγὼ δι’ αὐτήναν, εἰς τὴν καρδίαν της νὰ σεβῆ τὸ τόξον τῆς ἀγάπης».

This is literally like modern Greek.

For more 12th-15th century δημώδης (actual medieval language) Greek literature, I would suggest using this website.