The only hold that Chillingworth has over Dimmesdale is his hidden guilt and sin. Chillingworth has been torturing Dimmesdale because of his secret. Chillingworth knows that if Dimmesdale stands on the scaffold with Hester and Pearl, revealing the truth to the town, Chillingworth will lose his hold on Dimmesdale. This is why shortly after Dimmesdale dies Chillingworth dies. Chillingworth's entire being has been wrapped up in tormenting Dimmesdale for his affair with Hester. When Dimmesdale reveals the secret and subsequently dies, Chillingworth's purpose in life is lost as well as his hold over Dimmesdale.
In chapter 23, which is the end of the story of Dimmesdale and Hester, Arthur Dimmesdale already knows that his end is coming. He finally goes to the scaffold and he, in a sort of delirium, begins his last speech.
Chillingworth, who had a strong hold on his victim, noticed this and ran to prevent the clergyman from doing this.
"Madman, hold! What is your purpose?" whispered he. "Wave back that woman! Cast off this child! All shall be well! Do not blacken your fame, and perish in dishonor! I can yet save you! Would you bring infamy on your sacred profession?"
It is interesting to ponder Chillingworth's real concern. He obviously could have cared less that Dimmesdale falls to disgrace...but what about himself? He knew that he would be a collateral product of this disgrace. He would be the cuckold husband if he is found out. Chillingworth's disgrace would be equally proportional.
There is another thing to consider. Chillingworth made his sole purpose in life to suck the life away from Dimmesdale. This was to be done methodically, slowly, and coldly calculated. Having the prey leave before the predator has all his fun completely defeats the purpose.
That, however, is precisely what Dimmesdale does. He even gloats to Chillingworth that he beat him at his own game; that he will die, but not because of Chillingworth.
Therefore, what happens after Dimmesdale dies? Chillingworth dies shortly after. This demonstrates that Dimmesdale ultimately fueled the little life left on Roger Chillingworth; that their symbiosis was toxic, and that Dimmesdale cut it from its core by going to the scaffold to show his own scarlet letter.