Is there any advantage, when playing solo, to not having a companion fight by your side?
There are advantages and disadvantages. I'll outline them both below.
Advantages to going solo
- You can control what you aggro and when. If a follower gets too close, they can make that decision for you, often to both of your detriment.
- It will probably make you a better player, as you will learn not to rely on some of the crutches they can provide (some passive buffs, heals, etc.)
- Peace and Quiet. Spend enough time with a particular follower and you get tired of the same banter over and over.
Disadvantages to going it alone
- Not having them around will cause you not to benefit from the item bonuses they get (such as if you equip them with magic-find gear and the like) this will slow down your character development potentially.
- You will lose the benefit of some of their helpful buffs and support spells, like the Templar's "Inspiration" buff or healing spells. However, as noted above, this may help increase you personal skills.
- Some people like that sense of "not being in this alone." There are plenty of legitimate medical studies out there that prove sharing a mental burden has huge benefits. In this case, it probably can extend even to just having a digital follower.
- You do lose SOME dps, but as difficulty levels go up that becomes more and more marginal.
Interestingly enough, The original plan from Blizzard was to have followers lose all utility after normal mode was completed, however they have changed their stance recently and want them to remain viable throughout the game. They certainly provide a wide array of helpful skills, so I like to take them with me.
A companion will automatically attack hostile mobs, so it can be useful to not have one to control when you aggro an enemy(particularly Treasure Goblins).
You level faster if you go solo, as there is no one to steal your kills.