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Alternatively, the maximum modulus principle can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to open sets.
en.wikipedia.org The theorem may be interpreted to saying that a holomorphic function satisfying certain growth condition is necessarily algebraic: projective provides this growth condition.
en.wikipedia.org Informally, one sometimes summarizes the theorem by saying holomorphic functions are hard (as opposed to, say, continuous functions which are soft).
en.wikipedia.org This result has been generalized to normal families of meromorphic functions, holomorphic functions of several complex variables, and so on.
en.wikipedia.org This is an important restriction; if you start with a holomorphic function with an isolated zero, then any nearby holomorphic function will also have a zero.
en.wikipedia.org