No, it is not magical

Jeremy Crawford clarified this point on Twitter, using Sage Advice to support it. It is "background magic" similar to the breath weapon of a dragon.

Grant Myers @realgrantmyers

@JeremyECrawford ki is described as magical, but nothing in stunning strike says that it's magical. Can you confirm that it is not please?

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Jeremy Crawford @JeremyECrawford

Neither the Ki feature nor the Stunning Strike feature (PH, 78 & 79) is defined as magical for game purposes. #DnD

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Joe Lastowski @JoeLastowski Replying to @JeremyECrawford

Except in the section called "The Magic of Ki" at the start of the Monk description.

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Jeremy Crawford @JeremyECrawford

That is an example of the background magic I talked about in Sage Advice. Look for "Is the breath weapon of a dragon magical?"

Therefore, just as a dragon's breath weapon is not considered magical, a monk's ki is also not considered magical. They all work in an antimagic field.

However, Ki-Empowered Strikes are specifically magical

The feature says:

Starting at 6th level, your unarmed strikes count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.

This is also supported by Jeremy Crawford's tweet.

Jeremy Crawford @JeremyECrawford

The Ki-Empowered Strikes feature says a monk's unarmed strikes count as magical. That magic is suppressed in an antimagic field. #DnD

The Charm is not broken

The Antimagic Field spell explicitly states:

Any active spell or other magical effect on a creature or an object in the sphere is suppressed while the creature or object is in it.

The charming effect of the spell is on the target, not the caster. The antimagic field text says nothing about the caster. As long as the charmed creature stays out of the field, it will remain under the effects of the spell.

Jeremy Crawford rules similarly for concentration:

If you would like a more "word of the lord" answer, Jeremy Crawford said in this sage advice that:

Antimagic field has no effect on concentration. If it did, its description would say so.

All of this together makes it a safe bet that as long as the charmed creature stays out of the field, it will remain charmed.

Casters who die lose concentration

Note on PHB pg. 203:

You lose concentration on a spell if you are incapacitated or if you die

Death will cause you to lose concentration, but simply moving inside of the field will not.

So in the event of caster death on a concentration spell, like the examples provided, the charm will end. Regardless of if the caster is in the antimagic field or not.

Duration matters though

That being said, it is also confirmed by Jeremy Crawford that:

An effect created by a spell that's instantaneous isn't susceptible to antimagic areas or being dispelled. Additionally, if the caster doesn't have to concentrate on the spell, and then he later dies, the spell's effect will remain.

So if a spell causing a charm has a duration of instantaneous, the effect on the creature will remain, even if the caster dies. Additionally if the spell has a non-instantaneous duration, then the charm will be suspended if the creature moves into the antimagic field.

What about telepathic commands?

Some spells let you issue telepathic commands to the creature you have targeted. This is a slightly different effect. A link is defined in the oxford dictionary as:

A relationship between two things or situations, especially where one affects the other

Since the caster is on one end of this link, and the link is a magical effect (it originated from a spell), I would argue that the caster's link would be suspended while inside of the field, but the charm would not be. That is, the caster could not send requests out to the creature.

Would the caster realize he is in an antimagic field?

Hard to say for sure. There is no official ruling on this matter.

We already know the caster can continue to concentrate on the spells, so concentration won't help here. One of the answers in the related question suggests that because Antimagic Field states:

This area is divorced from the magical energy that suffuses the multiverse

a spellcaster, who frequently manipulates this magical energy, could potentially notice that something is wrong. But there is no mention in the rules, or in the spell text that this is the case.

The September 2016 rules answers clarify that a person would only know they were under a spell if there was some noticeable effect. If the charmer is aware that their target is no longer charmed, or that his telepathic commands are having no effect, then they would certainly know that something was wrong. But it seems like it's up to the DM if the act of intersecting with the antimagic field is a noticeable effect by itself.