= Solution
The partial-fraction decomposition is
$$
\frac1{1+t^2}=\frac1{2i}\left(\frac1{t-i}-\frac1{t+i}\right).
$$
The residues at $i$ and $-i$ are respectively $1/(2i)$ and $-1/(2i)$. If two paths from $0$ to $z$ have winding-number differences $n_+$ and $n_-$ about those poles, their <integrals> differ by
$$
2\pi i\left(\frac{n_+}{2i}-\frac{n_-}{2i}\right)
=\pi(n_+-n_-).
$$
Loops can wind around either pole independently, so the value changes by arbitrary integer multiples of $\pi$ and $F$ is multivalued.
Along the straight real segment,
$$
\int_0^1\frac{dt}{1+t^2}=\frac\pi4.
$$
Consequently the complete set of values is
$$
\boxed{F(1)=\frac\pi4+k\pi,\qquad k\in\mathbb Z.}
$$
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