I think Kogami did the right thing in killing Makishima
I know that some of you will go "Nyachakka" or something like that, considering I'm only now watching Psycho Pass.
But anyways, while I actually kind of respect Akane as a character (unlike so many other idealistic anime characters, who are just plain lame, insufferable, cringeworthy faggots who should just kys), despite all your capability, if all you're doing is practically giving an extremely dangerous criminal mastermind more and more chances to go and hurt, kill and maim people (while simultaneously giving something as downright evil as Sibyl what it wants, knowing it's true nature), then your legal idealism is ultimately cursing innocents to more suffering (not to mention the way her idealism was deliberately being manipulated by Sibyl into making her a pawn).
Now ofc, I realize that this sort of a judgment would be something far too obvious for the cold cynical bastards of inch, or from the POV of a person who is all about cold hard pragmatism, at an individual or collectivist level.
But notice I used the phrase "the RIGHT thing"'. Even if your fundamental motivations are idealistic, fuelled by a desire to protect people and be the good guy, as long as you're not utterly divorcing yourself from results, the decision is still justified (even if Kogami himself might've been partially at least motivated out of a sense of revenge .... although one could also argue that concept of "Justice" itself is a sanitized, contained form of revenge, exercised at a communal level ... however, I digress).
There will always be the need from time to time, when push comes to shove, for a Knight to rise up and slay the big, bad Dragon. Yes, SLAY. Not castigate, not convict, but SLAY.
While in Akane's dogged legal idealism, he may be a "murderer" (and also, I acknowledge that "doing what needs to be done" can be very much a slippery slope where eventually it ends up being used to justify legit bad stuff), I think Kogami is a bona fide hero for cutting the crap and simply putting down Makishima for good, then and there, instead of trying to be "the (morally) better man".
(Yes, I know ... "toh hum kya kare, bc" ^_^)