= Solution
Using complex coordinates near zero, $F_k(z)=z^k$ for $k>0$ induces winding number $k$ on a small circle, so its local relative homology multiplier is $k$. For $k<0$, $F_k(z)=\overline z^{\,|k|}$ has local degree $k$, since conjugation reverses the winding. For zero, take $F_0(z)=|z|^2\in\mathbb C$: its zero fiber is the single point zero, while the image of a small circle is a single positive real point, so its local degree is zero. Equivalently a nearby nonreal target has no preimages.
Each of these maps is proper and hence extends continuously over the one-point compactifications to a map $S^2\to S^2$ with zero as an isolated preimage of zero. Thus
$$
\boxed{\deg_0F_k=k\quad\text{for every }k\in\mathbb Z.}
$$
These are <local degrees of arbitrary integer value>. Taking $F_k$ times the identity on $\mathbb R^{d-2}$ gives the same local degrees for every $d\geq2$. In dimension one, an isolated local degree is restricted to $-1,0,1$, so the all-integers assertion is realized in dimensions at least two.
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