He is important, he's important to her and so many other people even if this isn't quite what they're used to doing. Staying in one place for a long period of time, not discovering the secrets in a short amount of time. Sometimes things just don't work that way. Like on Trenzalore.

At his question Clara loses the exuberant eagerness in her expression and something a bit like anger covered by neutrality kicks in. She is aware of the name, that the Doctor considered the Master to be his friend - his oldest friend but that doesn't mean she has to like that friend after all that's happened. Between Danny and the Daleks, Clara would much rather not ever see that particular Time Lord ever again.

She tries for casual when she does speak up. "Oh you know, befriending the Cybermen in order to get your attention, trying to not get killed by the Daleks. These days we're calling her Missy."

Then he's off and Clara's chasing after him without a moment to think it over. He isn't the Doctor she's traveled with, not for very long but he's still the Doctor and she can't give that up.