$\operatorname{Hom}(\Bbb C^\times, \Bbb C^\times) = \operatorname{Hom}(\Bbb R \times \Bbb T^1, \Bbb C^\times) = \operatorname{Hom}(\Bbb R, \Bbb C^\times) \oplus \operatorname{Hom}(\Bbb T^1, \Bbb C^\times) = \Bbb C \oplus \Bbb Z$