I... am a bit confused how they've made this connection of Hypatia of Alexandria as a defender of slavery. I've watched the show somewhat, but don't know this particular episode or scene, so I'll have to go with just what the title says, though there is not much to say about Hypatia's philosopy.

As far as we know, as a philosopher Hypatia was more of a teacher; there is no record of her ever writing original works of philosophy, and in fact not much any of her writing survives. Most of her literary work went into writing commentaries on mathematical classics, like on Diophantus' Arithmetica and Ptolemy's Almagest. Scholars believe that they have identified some of Hypatia's contributions in Medieval manuscripts, that compile mathematical commentaries from the antiquity, but this is all purely about high mathematics. (Which was in antiquity considered to be part of philosophy also).

We really don't know much about Hypatia's philosophical convictions, apart from the fact that she acted as a teacher in and later the head of Mouseion in Alexandria, the school of philosophy her father Theon of Alexandria had founded. From our sources we know that she taught her students mathematics and works of Aristotle and Plato, and more recent Neoplatonist Plotinus and Porphyry, one source describing her as "heir to the Platonic interpretative tradition handed down from Plotinus" (Socrates Schol. HE 7.15.). As this is pretty much all we have to go on, Hypatia's philosophical convictions can only be somewhat reconstructed from the basis of Alexandrian intellectual trends and the claim that she apparently specifically preferred Platonism as mediated through Plotinus, who was the founder of a movement we today title as "Neoplatonism" (although they themselves just thought of themselves as Platonists). We do know that Hypatia's students were Christian, while herself being pagan, so at least she didn't have a problem with religious tolerance.

In terms of how well Neoplatonism would stand up to today's mores, well... I guess nobody would have problem with it apart from considering it rather wacky, but Neoplatonism was never very concerned with formulating an ethical theory. They weren't necessarily very inventive and many of the ideas they promoted were derived from or very similar to e.g. Platonism and Stoicism. Neoplatonism was mainly concerned with metaphysics and cosmology, the main tenet being the mystical 'One' - this is really difficult to explain simply or, I think, understand even after some study. Basically, 'The One' is the first divine principle of reality, the one and only metaphysical source of everything. 'The One' is not really any substance or matter - it is too simple to even explain as something that does or does not exist. It is more like a sort of logic and rationale that is the source of every action and existence in the world, and ultimately everything that happens derives from this first principle. (Christian authors who were inspired by Neoplatonism later more or less equated the One with God, which is maybe the easiest way to try to grasp what they were trying to get at). Apart from the principle of One, the Neoplatonists fabricated equally convoluted theories of the nature of the soul, matter, and consciousness - they believed in the world-soul theory, namely that everything and everyone's consciousness in the universe is in fact part of the same soul "ecosystem" that cannot be divided. Neoplatonist metaphysics is not really considered as very important or influential in modern philosophy and is mainly studied as historical curiosity, although the movement definitely played an important part in the evolution of Western intellectual movements, such as in Christian theology.

In terms of practical ethical things, Neoplatonists advocated attainment of happiness and tranquility, which they believed was ultimately possible only through using all the human faculties of reason and philosophical contemplation. Internal, systematic reflection on universe and ourselves can lead to the understanding of one's true identity and Goodness, which should make people sort of immune to desires and external nastiness, and this is the definition of happiness. Plotinus doesn't really give much more 'pratical' advice on how to be a good human being apart from that. We don't know how much Hypatia interacted with Plotinus' student's Porphyry's work, who did do some more practical philosophy, by for example advocating vegetarianism - which I guess would be quite hip today. But, as far as I know, no Neoplatonists explicitly addressed the issue of slavery.

..I wonder if the writers got this idea that Hypatia was opposed to slavery, because there is a scene - and it's been a long time I've seen the film so I might be wrong - in the biographical film Agora (2009), where one of Hypatia's father's slaves sexually assaults her, and as a response Hypatia manumits him (i.e. sets him free and ends his status as a slave). I don't know if somehow this could be interpreted as Hypatia being 'anti-slavery', though I don't remember that Hypatia would have made any systematic stance about slavery in the film, I think the scene of manumission was more about Hypatia's non-violent way of getting rid of her attacker. (Manumission by itself is not really an 'anti-slavery' gesture in antiquity, people manumitted slaves they were close to/who had performed well all the time while continuing to keep other slaves.) Anyway, this scene is simply a fabrication of the film-makers, there is no such tradition of Hypatia being assaulted by a slave.