I think that people who at this stage think that Mihawk won't be connected to Imu at all are coping. I don't know if Mihawk will be Imus's son, a clone/experiment, or more an Ace and Whitebeard situation (obviously twisted), but some connection will inevitably be there.
Oda at least heavily implied Imus last name is Nerona, and had PERONA live with Mihawk for two years. He gave Imu the same eyes as Mihawk in the Manga. Yoru is pretty much the World government symbol turned into a sword. Mihawk's past is the one among the Warlords we know arguably the least about. Mihawk, despite his haters and even some of his fans claiming otherwise, is not going to be just some random guy who one day picked up a sword and became the best of the best.
Oda just does not write characters this way. Even Whitebeard himself had one of the strongest fruits ever and was likely a buccaneer.
The thing that always made me stumble is his last name, but to be real? I can see Oda going for some scenario where Dracule is not even an actual name but something that Mihawk picked up to not go by Nerona. It could be a Portgas D Rogue situation where he picked the name of his mother to distance himself from Imu, but alternatively what if Dracule is literally just a title?
The Crule, or Da Crule. Could be that Mihawk originally did not show promise as a child or that his mother either escaped while pregnant with him or hid him somehow before Imu could discover him. We know that nine years before the current story a terrible war was fought on Shikkearu Kingdom, the island out of all islands Mihawk settled upon.
He also carved a large wooden cross from a Tree. Could have been just to kill some boredom but it could also be where Mihawk's mother and her kingdom were laid to rest.
Lets say Imu/the Gorosei find out about a Nerona born on the blue sea. They straight up kidnap child Mihawk from his mother, who could have been the princess of this kingdom. Seems to make sense Imu would go for a highborn woman as his potential wife for his heir.
Mihawk grows up as a Heavenly Knight as a child, but either due to a lack of a proper rival (whom he would later find in Shanks) he eventually leaves and is able to return to his island. Perhaps Mihakw then discovers, his home had been wiped out in a war, potentially even a buster call.
That then makes Mihawk become the Marine Hunter. He goes on a crusade for revenge, maybe even kills the Gorosei that Saint Ju Peter, the youngest out of the five, ended up replacing. Mihawk is so cruel and indiscriminate in his revenge as he was taught to view everybody as an insect/rabbit to be hunted that eventually that gets lost in translation and somebody thinks Dracule is his actual last name. Or, more simply, it was the name of the royal family from Shikkearu Kingdom.
Personally would love that since it would make Mihawk a direct rebuttal to all of the Celestial Dragons "superiority" bullshit if Mihawk had been taught by them only to turn on them after deeming them all unworthy and below him in return/strike back in anger once he realized what happened to his home.