A Morse exhaustion function on the punctured plane I don't think I have a good intuitive understanding of the purpose of a Morse function. The classic illustrative example seems to be a genus 2 torus with the Morse exhaustion function being given by the height function. From that, I thought any point at which the topology of the sublevel set changes is "critical point". In that case, what would be the set of critical points of a Morse exhaustion function on the punctured plane (plane with origin removed)? I feel like saying that the origin must be a critical point, but surely we cannot say that, as the origin has actually been removed. 
If I could be given some help with correcting my intuition, I'd really appreciate it.

The problem with your example is that the punctured torus(or Plane) is no compact. The statement describing the topology of level set depend on those facts. For example important theorem fails:

Suppose $f$ is a smooth real-valued function on $M$ and $p$ is a non-degenerate critical point of $f$ of index $\gamma$, and that $f(p) = q$. Suppose $f^{−1}[q − \varepsilon, q + \varepsilon]$ is compact and contains no critical points besides $p$. Then $M^{q+\varepsilon}$ is homotopy equivalent to $M^{q−\varepsilon}$ with a $\gamma$-cell attached.

and another one

Suppose $f$ is a smooth real-valued function on $M$, $a < b$, $f^{−1}[a, b]$ is compact, and there are no critical values between $a$ and $b$. Then $M^a$ is diffeomorphic to $M^b$, and $M^b$ deformation retracts onto $M^a$.

This means that when $M$ is compact morse theory have powerful tools to describe the topology of the level sets of $M$. In your case $f^{-1}[-\varepsilon,\varepsilon]$ is not compact if the plane is the $xz$ plane in $\mathbb{R}^3$.
