Master Chief and Cortana’s relationship in Halo 4 is platonic.
Master Chief cares for her because he sees it as his job to protect her. Cortana feels the same way towards him.
“It was my job to take care of you,” says Master Chief.
“We were supposed to take care of each other,” Cortana replies. “And we did.”
The closest the two come to physical contact is a moment where Cortana, now being made of a substance called “hard light”, is able to actually touch Master Chief and not pass through him like she normally would, being a hologram. “I’ve waited so long to do that,” she says. What is Cortana’s reason for doing this? As far as I know a definitive answer has not been given, but here’s what I think: Cortana always had a mental connection to Master Chief; she was embedded into the suit of armor he wore, so it was like she was inside his head. But she never had a physical connection to Master Chief, and just once, before she died, she wanted to experience such a connection.
As Cortana drifts away, Master Chief says “Wait–” He doesn’t want to lose her; she has been his anchor, and without her he’s afraid he’ll be lost.
The relationship between Master Chief and Cortana is a rare example, not only in video games but in popular culture in general, of a non-sexual, intimate relationship between a man and a woman.