Bill Murray's performance in this film is quite notable as it indicated a bit of a career shift for him as he went from the sardonic lead of straight comedies to somewhat more dramatic work, although usually comedy is still involved in some way. The critics clearly noticed this and awarded this change giving him several critical wins for Supporting Actor for the year, but the actual industry awards all snubbed him. Although he was perhaps close to an Oscar nomination it is easy to see why he missed since Rushmore received no nominations. Also the two veterans, Duvall and Coburn, both clearly had a lot of support evidenced by Duvall's Sag win and Coburn's Oscar win. Harris was on a brief streak of being an Oscar favorite for a nomination, Rush was in a the best picture winner while having a banner year, and Murray's rival, in terms of the most wins from the critics, Billy Bob Thornton perhaps was the one who pushed him out.
Murray's performance here is one where he completely rids himself of any of that sarcasm that was one of the most notable aspects of his onscreen persona. Murray's role here is more defined by a somberness, and this approach is certainly fitting enough for his character Herman Blume. Blume despite being successful he clearly has a cold relationship with his wife, and also hates his obnoxious twin sons. Blume instead kinda reaches out to a potentially surrogate son with Max who he takes a liking to going so far as to financially support a few of Max's harebrained ideas. Murray is fairly good in setting up the emotional state of Blume as he expresses that longing in Blume for something he merely can understand. Murray makes it at least somewhat believable that not only would Blume go about helping Max, but he would also befriend him too.
Of course this still has comical qualities to it as is often the case with Murray. Murray is actually best, in the film anyways, when his scenes are not clearly done to be overtly comic. Murray is actually funniest here in just his quick insults to his sons or a few of his reactions to some of Max's insanity. Of course the film's problems comes into play as it almost seemed they weren't willing to quite let the old Murray go. This is in regards to the revenge scenes when Max and Blume come to a head over their mutual infatuation with Mrs. Cross (Olivia Williams) who happens to be a teacher at Max's school. This leads to some pretty standard slapstick comedy, that I did not feel Anderson did particularly well here, and Murray does not seem like he really has his heart in these scenes. It seems like maybe they want to call back to the old Murray, but Murray stays with the character, which is good, but doesn't make these scenes very funny.
This is a good enough starting point for the different phase of Murray but I would say it is merely a precursor for a far greater performance, that's The Life Aquatic With Steve Zissou for me a performance which the more I think about the more I like. Murray's work here I feel does hit the right notes as he should in portraying how his character feels with enough humor thrown into the mix as well. Watching the film again though I never thought any of it had that much of an impact. I simply never found his performance that moving or that funny. For example his breakdown at the end while watching the Vietnam play was well enough acted by Murray, but here it just seemed enough, nothing more. Now I'll admit this probably had a great deal due to the fact I did not connect with Rushmore as a film, and I do feel its weaknesses adversely affect Murray's performance. It's a good performance but it never became a great one for me.