Jason and Medea, by Gustave Moreau (1865).Public Domain, Courtesy of Wikipedia.
Not surprisingly, the 1963 Jason and the Argonauts movie looks very old-fashioned, particularly in photography and in what constitutes a buff male character*, although Ray Harryhausen's stop motion animation process special effects still look pretty good 46 years later. The film is worth seeing if the story of Jason and the Argonauts interests you, but not if you want the story of Medea.
Medea is brought in earlier in the movie than in the Argonautica of Apollonius of Rhodes, which covers approximately the same sequence of events. Chaffey's version's first sighting of Medea is when the Argonauts are still en route to Colchis. She is a beautiful, shipwrecked, damsel in distress. The character of Medea develops no further.
In the Argonautica, as in the movie, Medea is a priestess of Hecate. In the Argonautica, she is also the daughter of the king and the sister of Chalciope. In the movie, Medea sits back to watch Jason win the fleece on his own, in accord with a devaluation of women one would have thought more typical of the third century B.C., which is when Apollonius wrote his Argonautica.
Apollonius' Medea falls for Jason, thanks to the love god, but her decision to help Jason at the expense of her father is based partly on other familial concerns, specifically, the sons of her sister, Chalciope.
Jason, as a suppliant, recalling how useful Ariadne had been to another hero, Theseus [see Theseus and Ariadne], implores Medea to help him, in the Argonautica:
"I implore thee by Hecate herself, by thy parents, and by Zeus who holds his guardian hand over strangers and suppliants; I come here to thee both a suppliant and a stranger, bending the knee in my sore need. For without thee and thy sister never shall I prevail in the grievous contest...."
Medea then gives him a charm and tells him what he must do to fight the sown men who will come after him. [Argonautica Book III. 1026-1062.] Jason says he will take her away and marry her:
But if thou comest to those abodes and to the land of Hellas, honoured and reverenced shalt thou be by women and men; and they shall worship thee even as a goddess, for that by thy counsel their sons came home again, their brothers and kinsmen and stalwart husbands were saved from calamity. And in our bridal chamber shalt thou prepare our couch; and nothing shall come between our love till the doom of death fold us round."
The Argonauts win the sown men contest and then the next day Medea asks that Jason honor his promise to marry her and take her away. After Jason publicly promises to do so again, Medea takes him to get the fleece. Medea charms the monster guarding it, while Jason hangs back in fear. When Medea instructs him to, Jason grabs the fleece and they head towards the ship, where for a third time, Jason says he will take Medea home to become his wife. All this is missing in the movie, since its focus is firmly fixed on showing how great a hero Jason could have been.

(Hans) Sebald Beham (1500-1550): Hercules slaying the Hydra, 1545 (B.102, P.100 iv/iv) from The Labours of Hercules (1542-1548).
PD Courtesy of Wikipedia.

Comments
Thanks for the movie review! Why the “*” after “buff male character”? “devaluation of women one would have thought more typical of the third century B.C.” I thought that was primarily an Athenian thing. Was it true universally across Hellas? As to Medea’s motivations; I don’t think we can blame Love entirely. Plenty of people have loved without, betraying their father, dicing up there brother, sacrificing their children, burning a family to death in their home, tricking some girls into killing their father, poisoning a stepson and dethroning an uncle. Medea was just no damn good; from a mortal perspective. I read somewhere once that in black magic; power is attained through purposely breaking taboos. If that’s the case by the of the play (Euripides?) she a pretty power witch (or goddess) when she flies away in a winged chariot. Of course, if we determine she was a goddess, well that a completely different set of morals!
About the asterisk after buff male character: I’m not sure that people would agree with me, so I explained further. Along the self-doubting line, I also think, far more controversially, that Russell Crowe in Gladiator looked less than fit.
there is a modern Jason from The Netherlands set in the low tides country that was very powerful, brilliant and reveals how modern european women view the situation. Medea is stubborn and royal to the end when she does away with these two beautiful curly haired blond little boys in movie watcher traumatic fashion. I found it at the Bloomington, Indiana public library. Maybe they would provide the title and director, famous, upon request.