I wouldn’t compare GW Bush with Tiberius. Tiberius was known for his love of orgies. GWB seems a very decent family man.
I would say GWB is more like Claudius maybe? A stutterer, not well loved but in reality not a bad Emperor.
Claudius immediately followed Caligula, no ?
Okay perhaps Claudius then...but I say that W is still Commodus. Commodus, a failed emperor, once fancied himself a gladiator and actually went into the coliseum as one as a publicity stunt to curry favor with Romans. And I keep recalling "W" in a military flight suit on the USS Abraham Lincoln under "Mission Accomplished" when other presidents would have donned a suit and tie and rode to the carrier on a helicopter to visit the troops (Sailors).