Shin Okada said:“Lose" does not always mean death. Basically, nothing prevent a character from "firing" a cohort if he dare to take a bad reputation for mistreating or otherwise caprice.
Lose does not mean dismiss (or fire). It has entirely different connotations associated with it.
As I pointed out a familar can be replaced when it dies or is dismissed, so too can an animal companion. Both specifically state "dismissed" under the section for replacing. It is important to note that that word does not appear under replacing cohorts.
It does state that a mistreated cohort will leave.
"Mistreated cohorts become disloyal and eventually leave or even seek revenge against their employers. Loyal cohorts become trusted friends and long-time helpers."
Cohorts could also leave if there is a change of alignment that puts them in an opposition alignment to their leader. This is different than having a "different" alignment (there is a leadership score penalty for this one while it is flat out denied to have a cohort with an alignment in opposition).
Maybe I would have rather say, "replacing is the only way to continue having a reasonable cohort, as the cohort never catches up in relative character level to his leader."
And this is true when a cohort dies.