From “The Hour of the Dragon” (one of the best stories IMO, for being a full-length novel if nothing else):
the squires stripped Valannus of mail shirt, burganet and leg-pieces, and clad him in Conan’s armor of black plate-mail
Though plate-mail is arguably not full plate, it’s not “chainmail” either. Then there’s from Black Colossus:
At her command they brought harness to replace Conan’s chain-mail — gorget, sollerets, cuirass, pauldrons, jambes, cuisses and sallet. When Yasmela again drew the curtains, a Conan in burnished steel stood before his audience. Clad in the plate-armor, vizor lifted and dark face shadowed
And that, at least, is as complete a description of a full set of plate armor as I think anyone could need.
Then there are the mentions of knights in shining armor/plate-armor, featured in several stories, such as (again from Hour of the Dragon) this:
Valbroso rode down with his men-at-arms, a lean, dark man with
glittering eyes and a predatory beak of a nose. He wore black
plate-armor and was followed by thirty spearmen
Knights may not be common troops, but they’re not all High Lords either.
Sure. No argument there. Conan takes what he can get and doesn’t spend much time maligning his conditions - what is, is, and better “get on with it”.
It’s fictional, sure, but its tech-level is kinda all over the place, ranging from borderline stone/iron age like the Picts to high medieval like the Knights of Aquilonia, which probably rivals the 14-15th century [like you said] in our own (aka real) history. Then throw in magic and the whole “Advanced society we don’t fully understand (Atlantis etc) came before us” schtick for good measure, and you get the mishmash fictional setting that is Conan’s Hyborian Age.