TITLE: Are measurable homomorphisms $ (\Bbb{C},+) \to (\Bbb{C},+) $ or $ (\Bbb{C},+) \to (\Bbb{C},*) $ continuous, and do they admit an explicit description?
QUESTION [15 upvotes]: 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...

(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.

REPLY [5 votes]: I just thought that I should mention a result that considerably strengthens André Weil’s result mentioned by François in his answer above.
Adam Kleppner, in his 1989 paper Measurable Homomorphisms of Locally Compact Groups, proved that any Borel-measurable homomorphism between locally compact (Hausdorff) groups is continuous. It does not matter whether or not any one of the groups is Polish.