Vader had nowhere else to go, and no one else to turn to.
Family:
Father
He didn't know who his father was, or if he even had a father.
Mother
His mother, the only real family he ever knew, was dead.
Step-Family
He had only met his step-father and step-brother once, for a few minutes, and hadn't left on very good terms with them.
He had no family left, aside from a step-brother and step-father he had only met once for a day or so, and he didn't seem to care for either of them.
Friends:
Anakin had very few friends.
Obi-Wan Kenobi
Obi-Wan was certainly his best friend, but Obi-Wan had just chopped off both Anakin's legs and one of his arms, then left him to die on a volcano. That bridge had clearly been burned.
Padme
Padme was also his friend, but she was dead.
The other Jedi
The other Jedi had been allies and acquaintances, but he had personally killed dozens of younlings, participated in the murder of Mace Windu, and abetted the Emperor's plan to murder all the rest. The only survivors we know of are Yoda and Obi-Wan. We've already dealt with Obi-Wan; it is unlikely that Yoda would be any more conciliatory than Kenobi was.
Jar-Jar Binks
He might also have considered Jar-Jar a friend, but we don't know whether Jar-Jar would accept Anakin after all the atrocities he'd committed, or even how Anakin would have felt about turning to Jar-Jar after Padme's death. Speaking for myself, however, I'd rather die than spend five minutes with that infuriatingly annoying, blithering amphibious idiot.
He had murdered dozens of his friends, helped kill another, been mutilated by his best friend, participated in the plot to kill all the others - which was so successful that only two survived - and contributed to the death of his wife.
Places:
Tatooine
Anakin had spent most of his life as a slave on Tatooine, and grew to despise it.
I don't like sand. It's coarse and rough and irritating and it gets everywhere.
- Attack of the Clones
Coruscant
After that, he lived at the Temple on Coruscant. The Jedi who ran the temple were now dead, some by his hand, and the Temple had become Palpatine's palace. Coruscant is actually where he set up his primary residence as Lord Vader.
Naboo
His happiest days were spent on Naboo with Padme; returning to Naboo after her death would have been agonizing for him.
Of the places Anakin had spent a significant amount of time, none were particularly inviting now. He despised his true home - Tatooine, and he was too grief-stricken to revisit the place he'd been happiest - Naboo. On top of this, if he had chosen to abandon Palpatine after all that had happened, there wouldn't be a safe place for him to go - Imperial agents would hound his every step, an eventually kill him.
Resources:
Before his treacherous switch to the Dark Side, Anakin had shunned material wealth, as was expected of all members of the Jedi Order. The Jedi were supposed to own only some clothing, a lightsaber, and, perhaps a few small personal items. Anakin was somewhat unique in that he also owned two droids.
Real Estate
He had none.
Money
He had none.
He had no home to go to, and no money or valuable property with which he might obtain a home. He was faced with the choice of being penniless and homeless on his own, or being incredibly wealthy and powerful with Palpatine.
As per Wookieepedia:
Overwhelmed by his despair in the belief he had killed Amidala and their unborn child, which fulfilled his prophetic dreams of her death, Vader destroyed the medical droids and severely damaged the facility through the Force. He broke his bindings on the operating table and struggled to walk under the sheer weight of his new prostheses and armor. The only purpose that remained in Vader's life was his service to the new Emperor.
Conclusion:
If he left Palpatine:
He would have no one to turn to, nowhere to go, and nothing to support himself; Palpatine would presumably hunt him down and kill him for his disloyalty.
If he stayed with Palpatine:
He would become rich, powerful, have ample opportunities to use his skills for evil, and eventually, he would have a chance to avenge what his new master had done by killing Palpatine.
