Etymologically, and close enough to common usage, "complexity" is the degree to which a set of tools create further (pseudo-)tools through their combination. Literally, there's something more "folded up" among what's visible from the surface, allowing a greater number of stuff (on unpacking) to appear within a smaller area/glance. "Depth" is usually the kind of cognitive load that goes into deciding among those additional choices.
Let's take ARR Monk as an example. You had essentially 9 ST rotational weaponskills: Bootshine, Dragon Kick, True Strike, Twin Snakes, Snap Punch, Demolish; Impulse Drive, Fracture, and Touch of Death. (I'm ignoring One-Ilm-Punch because it only saw use on two pieces of PvE content in XIV's lifespan, essentially becoming a PvP-only ability. Impulse Drive was TP-inefficient but was for a time the way to put out maximum DPS.)
Ultimately, though, we don't play with different isolated abilities except in odd snap-decisions (and in XIV, rarely even that, due to combos or modifiers massively making one choice outweigh all others). Instead, we devote certain amounts of attention towards different considerations to be tracked/estimated and decide on sets of actions accordingly. To my mind, the extent/breadth of what all is worth tracking and making decisions from can generally be considered "depth"; our "complexity" comes from what discrete actions we can choose accordingly at the level we actually think about this stuff / put it all into action or optimization. The more interrelated those decisions are without being interchangeable (or inexpressive in their differences), the greater the number of actions ultimately available per a given situation, and therefore the greater the complexity.
In ARR Monk's case, before considering ToD and Fracture, we'd just have <Dragon-Twin-Demo-Boot-True-Snap; Dragon-Twin-Snap-Boot-True-Demo; Dragon-Twin-Snap-Boot-True-Snap> and a "Dragon-down" rotation at higher GCD speeds of <Demo-Dragon-Twin-Snap-Boot-True-Snap-Boot-True>. With them, though, a ToD had to fit into every standard string (every .83, really), which then improved the efficiency of that standard rotation by reducing Dragon/Twin clipping, and could perfect this via a Fracture per Demolish.
And because ToD and Demolish were basically not affected by Dragon, our number of good openers was thereby enlarged to include, especially after PB, the likes of Snap-Snap-Snap-Twin-ToD-Demo-Dragon-Fracture-True-Snap-Boot-Twin-Snap-Dragon-True-Demo and what drifting ToD per string, Fracture per Demolish rotations would come of it.
Add to that the fact that Demolish would offer a ~2-second longer GL (just 10s back then, meaning you only had ~3.5s spare per cycle, with Demolish therefore adding over half) because it'd snapshot its use of GL at actuation but wouldn't apply it until after the long animation had completed, and you've then got fight-specific openers to deal with uptime gaps.
So, a decent bit of depth (know what adds can be Demolished, know what jumps/gaps can be Demo'ed and how much you should overclock or underclock Demo's accordingly with ToD/Frac --or Impulse Drive, if it comes to that-- or an extra Snap, know how this affects your strings before it, know how those strings in turn affect your best opener) and complexity (the choice-making from whatever knowledge and/or tracking/estimation is rewarded).
Now let's compare that to Monk as is now. How much of a performance gap stands between completely optimal play and "refill orbs when they run out", and how much cognitive load and/or situational decision-making goes into that gap?
If I were to now say that EW Monk was more complex than HW Monk, how would you react?
Depending on the GCD tier, since each had different permissible rotations (and therefore swaps one could make / levers to pull), I'd agree.
How about Dragoon? Blood of the Dragon was a punishing mechanic in HW, made easier coming into SB, did that add complexity? Or was it more of a hinderance to the gameplay? How about HW Enochian? Similar concept to Blood of the Dragon, but even more punishing. We know the community at large didn't like these 2 as they were much easier to manage in SB.
For many, there would have been more "engagement with" BotD after its simplification than before for the simple fact that many would drop access to their Dragonskills (Fang/Wheeling) without that ever being a deliberate choice -- say, for eeking out one extra Gierskogul per BotD.
And while their tuning was initially set in such a way that you absolutely could blow a ton of Gierskogul casts and be happy even without them for the downtime (and I loved dungeoning with a high-SkS DRG that would let me do exactly that for high AoE and focus target damage, both, without every having to worry about needing a Bard), I've no doubt that BotD felt to many more like a wonky CD than like a core mechanic. Its flexibility (to burst and to wreck oneself, both) back then had both pros and cons, for sure.
HW Enochian, on the other hand, I can produce no defense for, especially for so long as the Elemental Timer still existed anyways. As much as it did add some varying intensity between having just recently hit it and having it ready to go again if one failed to maintain element, its "floor" was 95% of its "ceiling", with the remaining 5% being redundant with prior optimizations anyways. If all you do now was what you already did before, it'd can't be said to increase "depth" (unlike BotD), only punishment (and on something that already had plenty of punishment -- enough to actually reduce options for the smallness of their reward relative to that risk).