I am interested in generalizations of the following fact (known as **automatic continuity**, as pointed out below). I am especially looking for references to papers dating back to 1920’s. I feel that question like these have been well-studied when people were still interested in set-theoretic aspects of analysis...

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Cauchy) Any measurable automorphism $ (\Bbb{R},+) \to (\Bbb{R},+) $ is necessarily a linear function, and any measurable homomorphism $ (\Bbb{R},+) \to (\Bbb{R}^{\times},\times) $ is necessarily an exponential function $ x \mapsto e^{a x} $.

Is something similar true for homomorphisms $ f: (\Bbb{C},+) \to (\Bbb{C},+) $ or $ f: (\Bbb{C},+) \to (\Bbb{C}^{\times},\times) $ (assuming that $ \ker(f) = \Bbb{Z} $ in the latter)? (Yes, see answers below.)

That is, I am interested in facts which follow the following rough pattern:

If a map is not set-theoretically wild, e.g. it is measurable or Borel, and satisfies some identities, then it is in fact continuous, and, furthermore, it can be given by an explicit formula.