Do generalized Pauli Operators generate SU(n)? A commonly used generalization of Pauli Operators is the "clock" and "shift" operators summarized here. 
Pauli Operators are generators of SU(2). Are these generalized Pauli Operators generators of SU(n)?
 A: Yes, of course they do: J J Sylvester introduced them in 1882, p. 649, for that very purpose. They are a basis of all $n\times n$ matrices, and thus $\mathfrak{su}(n)$, the Lie Algebra of SU(n).  A linear combination of them with suitable coefficients, exponentiated, provides all elements of SU(n).
The complete family of n2 independent unitary (but non-Hermitian) independent matrices
$$ 
(\Sigma_1)^k   (\Sigma_3)^j  =\sum_{m=0}^{n-1}    |m+k\rangle \omega^{jm} \langle m| ,  
$$
where $\omega^n=1$, provides a trace-orthogonal  basis for $\mathfrak{gl}(n,C)$, known as "nonions" $\mathfrak{gl}(3,C)$, "sedenions" $\mathfrak{gl}(4,C)$, etc...
It should be evident that  $\Sigma _ 1 ^n = \Sigma _ 3 ^n = 1\!\!1$,
and the braiding relation,
 $ \Sigma_3 \Sigma _1 = \omega \Sigma_1 \Sigma _3 = e^{2 \pi i / n} \Sigma_1 \Sigma _3$.
Since all indices are defined cyclically mod n, orthogonality follows,
$$\mathrm{tr}\Sigma_1^j  \Sigma_3^k\Sigma_1^m\Sigma_3^s =\omega^{km} n~\delta_{j+m,0}  \delta_{k+s,0}~.$$  
This basis, not very popular for its lack of hermiticity, can be systematically connected to the standard Weyl-Cartan Hermitian basis. 
For instance, the powers of the basic clock matrix,  $\Sigma_3$, comprising the  Cartan subalgebra, map to linear combinations of the $h_k^n$ Lie Algebra elements. For n=3 ("nonions") it is easy to demonstrate their mapping to the Gell-Mann matrices and the identity.
More significantly, the n → ∞ limit is particularly transparent in this language, identifying $\mathfrak{gl}(\infty,C)$ with the Lie algebra of 
Poisson brackets.
A: Do you mean "generator" in the group sense or the Lie algebra sense?
As a group, the Paulis don't generate SU(2) since the Pauli group is finite.  The same is true for the generalized Paulis.
As a Lie algebra, the generalized Paulis generate SU(n) since $\{ \sum_{ij} \alpha_{ij} X^i Z^j : \alpha_{ij} \in \mathbb{R} \}$ consists of all Hermitian matrices.
