The function $T \colon V \to V$ is a homomorphism of abelian groups or equivalently $\mathbb{Z}$-modules. If we use Zorn's Lemma we can realise $V$ as a freely generated module over $\mathbb{Z}$ with generators $X$ say. Then $X$ is uncountable. Picking $X$ is like picking a basis of a complex vector space. You can define every $T$ by taking any function $X \to V$ and extending it to the whole of $V$ to be a homomorphism like you extend any map of a basis to be a linear map. Unfortunately I don't think there is any constructive way of finding $X$ so all this shows is there are lots of maps $T$.
Might these all be linear ? No as you can arrange them not to be. As $X$ is uncountable you can choose $\dim(V) + 1$ distinct generators from $X$. These must be linearly dependent so arrange $T$ to not be linear on them and then extend it to the rest of $X$ arbitrarily.
EDIT: Previous comment on automorphisms of $\mathbb{C}$ removed as not really relevant.