Let $X=\mathbb{C}^{2}-\{z_{1}z_{2}+1=0\}$ and let $\omega$ be the restriction to $X$ of the standard symplectic form on $\mathbb{C}^{2}$. One can easily check that the following map $f:X\rightarrow\mathbb{R}^{2}$ is a Lagrangian fibration:

$$f(z_{1},z_{2})=\left(\frac{|z_{1}|^{2}-|z_{2}|^{2}}{2},\,\log|z_{1}z_{2}+1|\right).$$

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The only singular fibre is $f^{-1}(0)$, which has the topology of a $I_{1}$ fibre. It follows that this fibration is conjugate to the topological fibration in Example 2.6.