It can be argued that Maxim's actions are certainly consistent with his character. As the other educator has already pointed out, Maxim is primarily focused on protecting his family name and reputation.
So, despite Rebecca's infidelity, he decides not to divorce her. In the story, Maxim tells the heroine (his new wife) about his last encounter with Rebecca. We learn that Rebecca often flaunted her lovers in Maxim's face. Accordingly, Rebecca used to entertain a supposed cousin, Jack Favell, at the Manderley estate cottage when Maxim was out of town. When Maxim found out about the dalliances, he was furious and warned Rebecca that he would shoot Favell if the latter ever crossed his path.
One evening, Maxim discovered that Rebecca was at Manderley again, and he suspected that Favell was with her.
I guessed she had gone off there then, down to the cove. And I knew then...
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I could not stand this life of lies and filth and deceit any longer. The thing had got to be settled, one way or the other. I thought I'd take a gun and frighten the fellow, frighten them both. . . .
I began at once about Favell and she listened to me without a word. "We've lived this life of degradation long enough, you and I," I said. "This is the end, do you understand? What you do in London does not concern me. You can live with Favell there, or with anyone you like. But not here. Not at Manderley." . . .
"You know the conditions," I said. "I've kept my part of our dirty, damnable bargain, haven't I? But you've cheated. You think you can treat my house and my home like your own sink in London. I've stood enough, but my God, Rebecca, this is your last chance."
Here, Maxim's words clearly demonstrate his anger at Rebecca. However, they also highlight his upper-class values and expectations about marriage. Maxim originally consented to a marital détente with Rebecca because he believed that the fallout from a divorce would sully his family's reputation. However, Rebecca's continued flaunting of her lovers led Maxim to reject his original resolution.
During their interaction at the cottage, Rebecca taunts Maxim about having another man's baby and passing the child off as Maxim's. Enraged by her words, Maxim shoots Rebecca, and she dies:
"If I had a child, Max," she said, "neither you, nor anyone in the world, would ever prove that it was not yours. It would grow up here in Manderley, bearing your name. There would be nothing you could do. And when you died Manderley would be his. You could not prevent it."
It can be argued that Maxim's actions here are consistent with his character. He will not consent to being both cuckolded and held indefinitely captive by Rebecca's schemes. So, he shoots and kills her.