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You may want to start here:

https://en.wikipedia.org/wiki/Group_homomorphism

The way I've understood it is that only addition has been possible so far. With a multiplication operation, you can get a field, and do "everything" with the real/complex numbers.

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Well a homomorphism applies to a group and a group can only have one operator by definition. That won't change probably. Rings have two operators, fields are rings that support cancellation and integral fields I believe support inverses as a rule and where you'd get "everything" I believe.

"Well a homomorphism applies to a group"

Not so. The notion's far more general: https://en.wikipedia.org/wiki/Ring_homomorphism

This gives us a notion of homomorphisms between fields.

https://en.wikipedia.org/wiki/Homomorphism for more general notions of homomorphisms.

You're right. My bad. Thanks for correcting me.

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