Consider a two-qubit Clifford gate that maps $|00\rangle$ to $|00\rangle$, up to a phase. Can it map at least one of the other computational basis states $|10\rangle, |01\rangle, |11\rangle$ into a nontrivial superposition of computational basis states (i.e. a linear combination of at least two states)? Or do no such Clifford gates exist?
Similarly, for a three-qubit Clifford gate that maps $|000\rangle$ to $|000\rangle$, up to a phase. Can it map at least one of the eight other computational basis into a nontrivial superposition? Or do no such Clifford gates exist?
The controlled-Hadamard gate maps $|00\rangle$ to $|00\rangle$, but it maps $|10\rangle$ to a superposition $\frac{|10\rangle + |11\rangle}{\sqrt{2}}$. This gate isn't a Clifford gate, but I'm wondering whether a Clifford gate could ever behave somewhat similarly, in that some computational basis states are preserved but others become superpositions.