What key would best complement a harmonica in the key of C?

Depends on what you are playing.
But if it is blues, get an A.

For a learner just C , later a set:

C D F G A

http://harmonicatunes.com/key-of-c-harmonica/

“Many beginning harmonica players want to play blues. As always, you need a harmonica which matches the key of each blues song.”

I like @Pinguidchance‘s answer that a set will be useful.
If the player is jamming with the radio etc they will need each of the popular keys.
Get harmonicas with metal reeds, the cheap harmonicas and sets have plastic reeds that start out sounding lousy and get worse.

@josie Thanls for the recommendation.

@Pinguidchance and @dabbler That sounds nice in theory. But a nice quality harmonica is probably around $80—$100. I can’t really afford to buy a whole set.

Hohner Marine Band Harmonicas 5 pack different keys. Costs about $150 for all five with case.

@Tropical_Willie I was looking at the Suzuki Hammond Promaster.

I’ve got both a “G” and a “C” Hohner Marine upstairs in the extra room that belonged to my father-in-law.

If the person is a beginner the promaster is way overkill…
A Hohner Special 20 is a great blues harp at a bit more than half that price.
The best harps I’ve used are Seydel Session Steel, with the stainless steel reeds, loud, clean-sounding, very durable – also cheaper than the Promaster.

I picked up a Hohner Special 20 as a Christmas gift for 9yo this year.

I’m looking at something nice enough to be part of a wedding gift for my niece and her fiancé.

The Session Steel is nice. But I’m not sure if they’re into blues necessarily, so I was looking more at the non-specific ones.

I’m also a fan of overkill! I like the idea of buying something a little more advanced that they can grow into and keep for a long time.

Session Steel would be great for any purpose because the tone is very accurate and stays stable way longer than most brass reed instruments. I have a couple in minor keys that I do not use for blues and they are bright and clear. I do usually use my major keys for blues and they are great for that, too.