I've been doing a season one rewatch, and I'm struck by how complete Liz and Max's arc seemed to be. Also, after the anti-hero panel at online Escapade Con, I think Max would actually fit into the definition.
The hero's journey, as defined on Wikipedia: a hero who goes on an adventure, is victorious in a decisive crisis, and comes home changed or transformed. I also like
how this site defines the hero's journey. Max's story doesn't follow that traditional model. Max doesn't go on an adventure (Liz returns to Roswell and kicks off the story) and the crisis is one of his own making - the past coming back to haunt him. In the end, he sacrifices himself for Liz and proves that he hasn't changed at all.
We start off with Max pining like hell. Liz feels attracted to him, but she doubts her feelings because of the handprint/psychic connection. Then she starts to suspect him of Rosa's murder, and that makes her nearly walk away. Max's dedication waivers when Isobel and Michael confront him, but then he learns Isobel betrayed him. Liz comes back to Max when she knows the truth and they finally share a kiss. They don't define their relationship, but they have sex in the season finale and seem blissful. Then Max finds out Rosa is in a pod, so he sacrifices himself to right one of the biggest wrongs he's ever committed. I'm convinced he knew the toll it would take, considering he was really messed up physically after saving Liz. He trades himself for Rosa, as a final act of love for Liz.
Liz's default is usually anger. She's not the self-sacrificing type, instead throwing herself into work to find a solution. Therefore, if Max had really died, I think Liz would have been angry Max didn't tell her about Rosa, and didn't try to find a solution. And I think Max knew that, but he thought it was still a fair trade.
The biggest take-away here is that Max would do anything for Liz. There are no limits, even if he knows Liz will disapprove. That means Max's moral compass is defined by those he loves, instead of doing what is 'right'. He does acknowledge this in season one, telling Liz he'll do anything for Isobel and Michael (when she makes that serum). Even though he's a man in uniform, he's not loyal to the uniform in the same way he is to his family. Max's morals are easily compromised to save those he loves, and I think this makes him an anti-hero.
It's easy to forget Max killed a random person pre-canon, because it seemed like he didn't mean to do it. But we learn from Noah that killing made Max feel powerful and satisfied. Max must have realized the true power he held at that point, if not before. And he never saved anyone else before Liz, because he had been wary of consequences. Liz makes him disregard consequences.
ETA: After a convo with infp on Discord, I'd say Max has a clear sense of morality, but he steps over his own line multiple times. Maybe that means he's got no line? Max is defined more by his emotions than by what's 'right.'
( Here are three of my favorite Max/Liz fanvids that capture their dynamic in s1 )And I completed RNM Comment Bingo and won a fic!
( My Card under the cut )