I'm actually not that interested in changing the Test captain. Mushfiq has done a pretty decent job in terms of using the DRS an he has the respect of everyone on the team. He is also the most consistent run scorer and can bat in all conditions so he leads by example. He also has a exemplary work ethic.
His major flaws are stubbornness and rigidity, and he has a tendency to blame youngsters and newcomers into the team even when it was obvious the fault lies with senior players. He is viewed as the reason why mediocre players like Mahmudullah (his brother in law) played about 10 Tests too many.
That being said I am a huge fan of the guy for his toughness on the field. He was struck on the body by a 150 kph beamer from Varun Aaron during the 2014 Asia Cup, shook it off, and still managed to score a hundred. He averaged 170+ on the recent tour of NZ while battling bruised fingers and finally stopped terrorizing the Kiwis when he was struck on the head by a bouncer. Ironically, he was widley ridiculed for crying in public merely because a BPL franchise owner publically called him a bad captain.
Tamim does seem disinterested out in the field. But he has been working hard on his fitness the past 2 years and its showed in his numbers.
Shakib was perhaps not ready for the job back when he had it, and maybe he could be ready now.
But my future picks for Test captain would be Miraz and perhaps Mosaddek, although I dont know much about the latter. However according to Tamim, Musa has a very sharp cricketing brain. That may be a relative statement.