Ok. To preface and put some general thoughts on the bosses entirely, here we go. I am going to go through each and every boss, and discuss them in an objective way for ranking. Hopefully, this will help the people who haven't played the game an idea of what these bosses even represent.
MASSIVE SPOILERS, DUH!!! I WILL BE USING THE FRUE NAMES OF CHARACTERS HERE!!! YOU HAVE BEEN WARNED!!! I WILL PROVIDE SPOILER NAMES FIRST IN LARGER FONT, SO READ AT YOUR OWN DISCRETION!!!
Bosses that Have a Strong Reason to be Ranked (a.k.a., the Ones I Really Think Should be Ranked Above all Others), in Descending Order
Bored Sociopath (Tohru Adachi)
The Plot: This is the most obvious rank. The character is present from the start of the game (He's like the 5th person you meet). He has large interactions with the party and the world, and is exceptionally important in understanding the story, especially enroute to the frue ending. I have the least to say about her, since there should be no concerns about him, based on plot.
The Good: No real interpretation issues (very straightforward). Relatively decent boss. Major plot player. Seen throughout the game, doesn't come out of nowhere.
The Bad: Respect might vary slightly. Possibly not as well-known as the next one, due to being on a technically optional ending.
Hippie Jesus (Taro Namatame)
The Plot: Ok, first relatively odd one. Hey-Zeus is present since the start of the game, at least in backstory. You know the first murder was related to him, and you get glimpses of him throughout the game. This culminates with the 2nd climax of the story, when a character is kidnapped. As for his boss form, it is him. Hey-Zeus's boss form is called Kunino-Sagiri, the Goddess of the Fog. It has his same voice, attitude. Building it into the entire plot, where Ameno-Sagiri is the God of Fog and the direct result of Cabbage Man's push to annihilate the world, Hey-Zeus is the subservient (hence Goddess instead of God) - indirectly, he is assisting Cabbage Man in creating the fog. The boss form is not his Shadow - it is him. One of the stated powers of the persona is that it can manifest the way you want it. Fueled by Hey-Zeus's desire to "save" people, he becomes a "Messiah", with a halo and giving a peace sign. Shadows, as stated, ARE the Persona of a person, in it's most basic form, and Hey-Zeus doesn't let them take over him - he clearly knows after battle what he did. He used the power of the persona to become what he thought he was - it didn't overtake him, he clearly was using it and in perfect mind ("I...I'm going to save her! Don't interfere!"). There's also a huge infodump after the fight, and you realize he was a bigger player than you thought - things do piece together, and he's vital in deciphering who the real villain is.
The Good: Memorable boss fight. Relatively important player. The ONLY non-Shadow boss on this list that everyone faces (so he has the MOST exposure, no matter what), so if you're looking for major plot bosses with the most exposure, this is him. HIPPIE FREAKING JESUS.
The Bad: Respect varies a bit, and doesn't probably translate as well as you'd expect. Not as present as Cabbage Man is in the plot.
Arkham Asylum Inmate (Izanami)
The Plot: The orchestrator of the game, Deadite is very akin to Beatrice in Wild ARMs 3. You meet Deadite as the 4th person in the game, and you also see her in your dreams (the fog battle near the start of the game - you see the outline of him). Talk to the Attendant, and he'll actually provide hints on the identity of Deadite. She is alluded to throughout the game. She is the most obscure major boss, but easier to find than Loki without a guide. Not too much to say else, as the plot is generally straightforward here.
The Good: Strong boss. Important, final boss of the game. Relatively interesting for a Godlike boss, and generally not broken. Very straightforward boss to interpret, overall.
The Bad: Is relatively obscure compared to the other two. May be construed by some interpretations as too powerful.
Bosses that have a good case for being ranked, but are not as important as the aforementioned ones, and thus, can be done without
Visine Commercial (Ameno-Sagiri)
The Plot: More of a mechanism than a player, Ben Stein was created secondary to the desires of a certain guy with a love for cabbage. Ben is NOT a part of a crazy Goddess, but a being created to carry on the will of the "test subject". As noted, the plot on Ben is not very deep for it itself, though it is important to the story. It is more of a summoned force, akin to Cloud of Darkness (in fact...very much like Cloud of Darkness).
The Good: Great, interesting boss (loses to PCs, beats bosses, in Godlike!). Exceptionally memorable fight.
The Bad: May be construed as too powerful. Not extremely big on plot throughout the game. Not a part of the required ending, so seen less than the Hippie.
Bosses that I honestly don't feel strongly about, but won't be exceptionally irked if they make it in
Fetus (Shadow Mitsuo)
The Plot: Fought at the height of the first major climax of the game. I'd like to state again that Mitsuo and Mitsuo's Shadow ARE separate. Mitsuo rejects his shadow, unlike the rest, and they are clearly different (similar to the PC shadows). Not much else to say here.
The Good: Exceptionally memorable and interesting (as Djinn said, a great mascot of sorts).
The Bad: Not a particularly important plot person (the Shadow...the character, Mitsuo, is...barely relevant early, kind of mid-game).
Bosses I have no really support for ranking
The Other Shadows (Shadow Yosuke, Chie, Yukiko, Kanji, Rise, Teddie, Naoto)
The Plot: Clearly, the Shadows are GONE after the fights, becoming the character personas (as the plot states). These are more akin to dungeon bosses than major plot bosses. While very interesting, and generally very memorable (except for Shadow Yosuke and Chie, the two here I would be the absolute LEAST in support of), it's very akin to ranking, say, all the Brionac members in Wild ARMs 4.
The Good: Interesting and generally memorable.
The Bad: Not heavily important to the plot.
So, where do I THINK we should go with these?
The most important ranks: Sociopath, Hippie, Inmate.
Good, but not necessary rank: Visine
Ok, but not necessary rank: Fetus
Eh, probably not good to rank: The other Shadows
So, I meant to get this out earlier for people to look at, but my internet crashed around midnight ;_; I hope that helps.
Anyway:
Persona 4 Bosses
Rank: Sociopath, Hippie, Inmate. Abstain on Fetus and Visine (may change later). No to the other bosses.
Wild ARMs XF
Rank: Clarissa, Felius, Labyrintha, Tony, Levin, Ragnar, Alexia, SAMUEL L JACKSON, Rupert, Eisen, Chelle, Piedras, Charlton, Weisheit, Fear Clymsian
Boot Asgard (Bluelike, as I have heard everyone say)