More speculation: I don't think he was looking for Chief at this point, rather, trying to glean information about him from Halsey. Or force some sort of admission from her.
This part stands out especially, when you listen to the intonations and reaction:
https://www.youtube.com/watch?v=GbjA6rB98Yk#t=135
He challenges her, almost angrily when he says the word broken - and immediately Halsey recognizes something weird about that question, and deduces his intent.
He says the Master Chief is dead... seems almost sombre-like.
Now in the timeline, at this point, everyone believes Chief is dead - even this character. So when Halsey says he's MIA, he sort of challenges the adage of "Spartans Never Die" in a pretty condescending tone.
I think he wants to be the "next" Chief. The new beacon of humanity, the celebrity. He seems almost jealous, as if trying to specifically discredit Chief (at first seems like he's trying to discredit the Spartan II program, but his fishing tips Halsey off). This could also just be her first finding out about the Spartan IV program, and this guy is one of the designers.
Now Chief coming back during Halo 4 makes it kind of weird... though he didn't have all that much down time, really. Wild theory? This Agent Locke guy tries to kill Chief after Halo 4, when he goes to investigate New Phoenix, hence the broken armour and cloak in the reveal trailer. Maybe under instruction of Osman at ONI, maybe under his own volition. Chief takes off after finding something... maybe a message from someone. I guess that next Escalation comic vol. 8 would debunk this.
Arbiter seems to not trust the guy, based on what he says in the teaser, but he also seems to think that he is the only hope in finding Chief. Arbiter is responsible for "turning" Agent Locke.
Halo 5 Guardians. Chief, Arbiter, Agent Locke.
/wildspeculation