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Someone who had more than two facial expressions, (ie, could actually omg act), looked masculine enough to be credible rather than a pretty boy, didn't look like he should be in school (seriously, if the Doctor gets much younger, his female companions will be accused of cradle snatching!), and who could project some on-screen charisma and a bit of 'edginess'--the whole 'I'm not human and I'm seeing the bigger picture here'.
That said, I have no candidate. I've been a Dr Who fan for many years and at this point, I'm heartily glad to be shot of the welsh idiot who thought he could write and turned the series in a trashy soap opera. Maybe now we'll get some decent stories; credible forward-projected science, plot holes tied off neatly, convincing threads, gradual authentic character development, and if you want to create a story arc, plot it out from start to finish and then slip bits in to each completed story.
But I won't be holding my breath...