If the set of factors of an infinite word $\xi$ is regular, is this property stable under "shift's" of $\xi$?
Let $A = \{0,1\}$ be a binary alphabet and let $L = A^* -\{0,1\}^*00\{0,1\}^* = (01,1)^*(\varepsilon + 0)$. I denote by $F(w)$ the set of factors of a word $w$ and by $P(w)$ the set of its prefixes.

Let $w$ be an infinite word such that $F(1w) = L$ and $P(1w)$ is nonregular. Then $F(001w) = L \cup 00P(1w)$. Since $L$ is regular and do not intersect $00P(1w)$, $F(001w)$ is not regular. You can use Yuval's approach to give an explicit construction: take an enumeration $u_0, u_1, ...$ of $L$ and consider a word $1w = 1u_01^2u_1 \dotsm 1^nu_n \dotsm$.