Interesting. I tend to think of complexity in terms of "how hard is it for a total beginner to learn?" because that's my context for D&D Club.
In my experience, at low levels it would be:
Artificer: High
Barbarian: Low ("I would like to Rage.")
Bard: High
Cleric: Average
Druid: High
Fighter: Low
Monk: Average
Ranger: Average
Rogue: Average
Sorcerer: Very High
Warlock: High
Wizard: Average
Sub-class choice will obviously have an effect whenever those come online. A champion remains low difficulty, but a battle master is high.
What typically makes a class harder for beginners is having to learn multiple systems. So even though I tend to think of, say, a warlock as. simple class, evocations are actually very hard for a new player to choose and understand because they have no context. They don't even know that Eldritch Blast will be their bread and butter spell unless I tell them. And sorcery points are the bane of my existence with most new players who pick sorcerer.