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It is tempting to write $|\psi\rangle \in \mathbb{C}^d$ for a qudit state, but this isn't very precise because of global phases.

What's a better notation for the set of states of a qudit?

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The qudit Hilbert space is $\mathbb{C}^d$. But, as you say, the Hilbert space does not exactly correspond to qudit states because of global phases (and also the normalization requirement). The object which does correspond to qudit states is the projective Hilbert space $\textbf{P}(\mathbb{C}^d)$, which is equivalently the complex projective space $\mathbb{C}\textbf{P}^{d-1}$.

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