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[Star Wars: Episode III] What if Obi-Wan truly underestimated Anakin's power on Mustafar?

At the end of the climactic battle Obi Wan stands atop the high ground, giving him a major tactical advantage which he used to mutilate Anakin. But what if Anakin had jumped a bit higher? What if he immediately cuts his old master down in a swift move of his own? What if they continue their fight but it ends inconclusively with both alive and relatively unharmed?
gpt
So Anakin defeats Obi-Wan, goes back to his family, and returns to the Empire with them?

Anakin, victorious, would see his power grow exponentially as he continues training with Palpatine, and within a few years would be strong enough to depose his master, as is the way of the Sith.

Not only is Anakin a war-hero and one of Palpatine's most trusted confidants, he also uncovered and defeated the Jedi plot against the Republic/Empire , so the Senate and the People believes him when he tells them Palpatine died peacefully in his sleep and they don't mind when he tells them he will assume office of Emperor immediately. 

Assuming that Padme lived, she would be under constant surveillance, and be routinely manipulated by a paranoid Anakin into submission through the Force. She would, however, be defiant in her moments of freedom, and use her position in the Senate to protect the weak and safeguard the remaining  democratic institutions. Anakin doesn't mind so long as she doesn't touch the army, he very much prefers to spend his time with the 501st, squashing one rebellion after the other, hunting surviving Jedi here and there, and meditating on the Dark Side.

Anakin, while supreme commander of the armed forces, is quickly becoming more and more a beloved symbol of the unity and strength of the empire, rather than an actual ruler. The Senate rules instead, as it has always done, though Padme has become the de-facto Chancellor through her association with Anakin - some otherwise openly rebellious Senators, such as Mon Mothma, are instead convinced by Padme to work towards a balance within the system for the sake of peace.

Anakin has become the only remaining Sith, though because his training into the Sith-ways was not very comprehensive before he grew powerful enough to depose his master, and he remembers his previous training very well, he doesn't really care about the order as a whole very much. Coupled with the fact that his life is pretty good altogether, with family and a peaceful galaxy and all that, he lapses from a moderate Sith into something entirely different over time.

When they become old enough, Anakin would start taking his children with him in his campaigns, training them in the ways of war and the force. The children absorb the knowledge enthusiastically, though what they are becoming is not Sith in any traditional sense and they are definetly not Jedi.

At one point during his campaigns, Anakin would have found Yoda, and the two would have a climatic fight that the Emperor would ultimately win. Most remaining Jedi eventually perish at the hand of the Skywalkers, though many would convert and join the Emperor.

Eventually, Anakin would die, probably in some far-flung battlefield in the Outer Rim or something like that. By this point, both the Skywalker children would be grown adults and would have had plenty of experience in the military, some would fear a civil war between the twins, but nothing of the sort would happen - Padme's work in the senate has come to fruition, Imperial law states the office of Chancellor would be split from the office of Emperor upon the death of Anakin, with the Chancellor elected by the Senate, and the eldest male child becoming the new Emperor.

Luke is acclaimed Emperor by the senate and a thousand systems, while Mon Mothma is elected Chancellor a few months later. Princess Leia is unsurprisingly fine with all that, she never fancied to rule, was always more like her father in that regard - no, she has other plans...  

...within a few years, Princess Leia has gathered and organized his father's teachings about the Force into coherent whole, with the intention of starting a new order of force-sensitive knights with the purpose of protecting the peace in her brother's Empire. The members of the new order are nothing like anything that previously existed in recent memory, they make no distinction between sides of the force, and make use of the full force in balance... just how Emperor Anakin eventually came to use the force, unknowingly.

Thus, Anakin's empire and death has ushered a new age of unprecedented peace, and by dying both as the last Jedi and the last Sith, he has finally brought a certain old prophecy to a close.