Star Wars draws a new parallel between Vader and Dooku
In the season 2 premiere of The Bad Batch, Star Wars made a convincing comparison between Darth Vader and Count Dooku after Order 66.

Star Wars: The Bad Batch makes an interesting comparison between Star Wars villain Count Dooku and Darth Vader. Season 2 of Bad Batch recently premiered, continuing to explore the aftermath of Order 66 and the growing power of the Empire. In the meantime, the Empire is dealing with loose threads left over from the Clone Wars, including but not limited to the rogue Clone Trooper 99 aka The Bad Batch. After Palpatine manipulated Anakin Skywalker to kill Count Dooku in Star Wars: Episode III - Revenge of the Sith, Count Dooku's legacy and vast wealth on the planet Serenor is Another such loose thread that further cements Anakin's lineage as Darth Vader.
Although the Bad Bunch hoped to reap a small fortune from Count Dooku's spoils, this was not the case, as they were soon hunted down by clones working for the Empire and forced to abandon their mission. However, in the Star Wars: The Bad Batch season 2 pilot episode "The Spoils of War," the chase did enter Count Dooku's throne room. However, in Bad Batch, unlike its predecessor Star Wars: The Clone Wars, Dooku's throne room is very clearly visible, a piece of furniture, in In particular, Darth Vader and Dooku's respective roles as supporters of the dark side of the Force are compared.
Dooku's Throne Has A Very Familiar Sith Design

The design of Count Dooku's throne in his palace on Sereno is very similar to Darth Vader's towering fortress on Mustafar. Not only did Vader choose to inhabit the planet most closely associated with his fall to the dark side after his confrontation with Obi-Wan Kenobi in Revenge of the Sith, but Mustafar is also an important part of Sith history. Among some Star Wars tie-in media investments, most prominently the comic books, explain the towering, pointed design of Vader's fortress as a way to amplify Mustafar's Force, as Vader builds his castle in the dark face-to-face contact.
The source of the dark side on Mustafar is an ancient Sith cave, deep within the planet's fiery surface. So, with Darth Vader arriving on Mustafar, and darkness and hatred already spreading across the planet, the dark side of the Force has become incredibly powerful there. Vader even built his infamous red lightsaber on Mustafar after killing a former Jedi to get their weapon and most importantly Kyber crystals are inside. Pouring all his hatred, pain and sorrow into the crystal, with the help of the all-encompassing and amplified dark side forces on Mustafar, he was able to make it bleed and build his infamous red lightsaber.
Like Vader, Dooku Must Have Been Based On A Force Vergence

Considering that the shape of Dooku's throne is very similar to Vader's fortress, it could be interpreted that Dooku's palace was built on a similar concentration of Forces as Mustafar. If that particular design was meant to enhance the power of the Force, it's no wonder Dooku decided to use it. When he returned to Serenor after rejecting the Jedi Order and reclaiming his inheritance, he may have sensed the power lurking beneath the palace and worked out a way to amplify it for personal gain. After all, Dooku was Darth Sidious' apprentice at the time, ready to use any powerful advantage to prove himself.
Of course, Dooku and Vader have much more in common than their palaces and the designs they use to enhance the power of the Force. Both were manipulated by Palpatine, used and discarded, all in the name of Palpatine's quest for ultimate power. Both were disappointed with the Jedi order, both are forced to grieve someone they once cared about. Vader is just a more powerful replacement for Dooku, but it's interesting that Star Wars: The Bad Batch will draw new parallels between these two villainous and tragic characters, especially if something like Architecture simple stuff.