The game definitely wasn't expecting the order in which I completed this first quest.
After getting the Microchip, I went and got it implanted at the Hospital without first stopping at the Aegis Camp. When I got back there, Oldfield was asking me for the Microchip, and I could only tell him I didn't have it. Someone was blocking the panel so I couldn't interact with it. I casted a Psi spell, which briefly got him to move, so I could interact with the panel, which got us inside.
Inside, the dialogue was still clearly bugged. Oldfield asked me for the chip again,and talking to Briggs shows the same thing too.
https://steamuserimages-a.akamaihd.net/ugc/777357465303282001/081E4356BC67087C7394FCDF70C0A3C54981F2FD/https://steamuserimages-a.akamaihd.net/ugc/777357465303304762/B4B153F640D17FCBE499B709A7A9BD52C0B20A62/https://steamuserimages-a.akamaihd.net/ugc/777357465303305248/DFB6BDF3534E990E674C3381F2F5BB2AD17AB433/EDIT:
I just went back and had the chip removed, but Oldfield is still asking me for the chip, which I can only say I don't have. I think getting the XP and the quest log update at the Hospital is what's causing the problem. I don't think I can finish the DLC now.
https://steamuserimages-a.akamaihd.net/ugc/777357465303348679/0328C5F13F9532DF1EB3E1DE3A49F8BC746B371E/