My Mum always compares the Twelfth Doctor to the Third Doctor, but personally I don’t feel Peter Capaldi’s incarnation becomes like Jon Pertwee’s until Series 10. Series 8 he’s much more cold and distant, and I find it hard to imagine Pertwee’s Doctor abandoning Clara on the moon in Kill The Moon.
I think she has a point regarding how he is in Series 10 though. His grandfatherly relationship with Bill is not too dissimilar to the Third Doctor’s with Jo, and the dynamic between 12 and Missy is similar to 3 and the Delgado Master, given the strong emphasis on their history as friends.