Spock had a hard time managing his emotions. He knew Jim was someone who he respected, who he held in high regards, but he never thought of him as a friend, that is, he never admitted it to himself. When he watched him die, something had snapped inside of him. The emotions that he spent so long trying to repressed swam their way to the surface into a bubbling rage. He did the first thing he could think of - he went for revenge. It was illogical, but at that point in time he didn't care. He wanted Khan to pay. For some reason alien to him he wants Khan to pay, and he didn't necessarily care how he did it.
When Uhura first came down and told him to stop, he didn't want to. He wants to continue bashing in Khan' s brain until he felt the same fear Jim did right before he died. But then he heard that they could save him, that there still was a chance, so he took it. And it paid off. It had been a tricky first few days, but soon Doctor McCoy was able to tell him that Jim has a fighting chance and he had felt a weight lift off his shoulders. He was finally able to think rationally again.
He visited Jim a lot during those days. He wanted to see with his own eyes that he was okay. Logically he knew that McCoy would tell him if something happened, but he wasn't thinking logically. After Jim first woke up, his visits lessened. He was in the hospital cafeteria when he ran into McCoy a couple days after Jim woke up.
"Doctor." Spock said seeing him checking out with a coffee in his hand.
McCoy looked up and smiled. "Didn't expect to see you here, Spock." Street setting the confused and perhaps betrayed look on Spock' s face, he backtracked. "I thought Star Fleet would be having you buried under a pile of paperwork."
"Thankfully since the captain has made his recovery I only have to do a few pages worth." He stated matter of fact making McCoy smile.
"Now Spock, was that an attempt at a joke that I just heard?"
"An attempt, yes."
McCoy put his arm around the man's shoulders. "I do believe," he said smiling, that our young captain is beginning to influence your sense of humour.
McCoy led Spock back to Jim's room. "You can stay with him for a while of you want. I've got to go home for a little bit. I haven't had a shower in a week."
Before Spock could even protest McCoy was out the door. He walked into Jim's room and found him asleep. Spock uncomfortably sat in the chair next to him. Seeing Jim like this brought up memories of him dying. Shock focused on the rise and fall of his chest.
Spock had to admit something, something he hoped that when he asked would make him return to thinking logically. "Captain... Jim." he began. "I believe that I have failed you... as second in command, and as a friend. I have not had many friends, but I swear to you that I will try to do better. We may not have always been friends, but now I shall forever be yours." Spock looked satisfied with what he said and looked down at Jim again. "Now if you'll excuse me for a moment captain, I do have some paperwork I need to fill out. "
If Spock hadn't of been so busy trying but to focus on Jim, and instead work on his work, he might have seen the young captain's lips turn up in a tiny smile before going normal again.