"아름다운 모자를 사주세요!"
Translation:Buy me a beautiful hat!
Yes, -주 comes from the verb equivalent "to give" and adds an air of service to any verb, some "for someone's benefit". So if person A does something in favour of person B (which is mostly the speaker), you ought to add -주- to the -아/어 form of the actual verb. Now this does not necessarily mean that the recipient of the beneficially intended action needs to be the first person, but it mostly is. In other cases you will usually encounter object nouns or pronouns (or reflexivity elements) somewhere before the verb for the sake of clarification (if not provided by context anyway). So if it told you you were wrong in choosing a different variant, duolingo is wrong.
Example: "도와주세요!" -- "Please help!," which would most likely translate to "Please help me!" in any pro-drop language and almost all others I know. But it can also take "Please help him!" or other meanings. So the most frequent reading would be "Please help me!," but the others would be covered as well.
(A bit irregular, for 돕다 becomes 도와 in the -아/어 form.)
i think that as much as we can try to translate the 주다 presence in the verb, we can't. As far as I know, the 주다 here gives the notion of "as a favor", like "(please) buy a beautiful hat for me, as a favor" but the "favor" part is already implicit in the translation when we add the "please" It's something that we should have in mind, but isn't translatable (i think)
(With Korean each time I go for a literal translation, it turns out just the variant I have not picked which is required. I know it is hard to design functional language courses, it happens in basically all other languages as well (more advanced and L1 options are generally not considered valid as much as poor L2 smattering), and I know Korean is still in a beta phase -- but I strongly recommend introducing a general synonymity database to avoid such frustrations. I would say my false negative quota exceeds 20%. Even though I like the gamification aspect of duolingo, it is lacking in functionality literally everywhere, even in my native tongues. It seems to educate people to speak languages brokenly and to avoid any elaborate speech. I would even like it better if I could avoid those "false negatives" test methods altogether. Just as a general remark.)
And: Yes, I went for "Please buy me a beautiful hat" here. Which is more corresponding than anything else around. I would even say that the ending does not even allow for a more impolite translation (as the one presented).