I last played enchantment about 5 or 6 years ago before most of the supplemental material was published.

Since all the new books came out I really like to  multiclass and play unique characters.  Most of the wizards I have played recently are multiclassed - typically  bladesingers with a dip.

I think I want to try an enchantment wizard again but I am having trouble with a multiclass pairing.  I really like the idea of using hypnotic gaze.  Fail one save and an enemy is out of the fight as long as you stay within 5 feet AND continue to use your action.  With no save after the first it is pretty powerful BUT it takes the wizard mostly out of the fight too.

So I am trying to think of some pairings with it, here are a few ideas but I am luke warm on most of them.  Overall this is a level 3 to level 12 level campaign:

1. Level 4 Battlemaster -

Pros: heavy armor.  Could action surge and put down a control spell, then hypnotic gaze and finally get a quick toss so he can still deal damage with a bonus action.  Get superior technique for an extra die and maybe martial adept as well to get 6 per short rest.  Pick up sharpshooter and use darts.

Cons: With only 1 attack damage is not great compared to full martials.  4 lost spell levels

2. Level 4 Arcane Trickster

Pros: Only 1 spell casting stat.  Lots of bonus  action options while using hypnotic gaze, some limited bonus action control with mage hand legerdemain, extra cantrips .

Cons: No sneak attack while using hypnotic gaze, 3 lost spell casting levels

3. Level 4 Eldritch Knight

Pros: Heavy Armor, extra cantrips, could action surge to uuse hypnotic gaze on 1st turn does not use as many spell levels as another fighter.  

Cons:  After action surge pretty much useless while maintaining hypnotic gaze. 3 lost caster levels

4. Level 6 shadow sorcerer

Pros:  No lost caster levels, lots of extra cantrips, quicken spell while gazing.  Hound of ill omen.  Can be done with a 13 or 14 charisma

Cons:  limited sorcery points, 3rd level spells max, difficult thematically.

5. Level 4 Fathomless Chainlock (sprite familiar)

Pros:  Invocations, at will bonus action attacks with sprite or tentacle.

Cons:  Heavily MAD.  Loses 4 wizard levels.  

Thematically, I'd work backwards and figure out the flavor you want for hypnotic gaze, and let that guide the choice of the other class

Mechanically, sorcerer with a Metamagic Adept top-up feels like the best option at first glance. Being able to quicken spells while maintaining the gaze would be sweet, and it's the only combo you listed that doesn't lose any casting potency. Also, Eyes of the Dark plus Hypnotic Gaze seems like it would be a super-fun/super-creepy combo, since the target only has to be able to hear you for it to work...