Summary: We consider a generic family of polynomial maps \(f := (f_1, f_2): \mathbb{C}^2 \rightarrow \mathbb{C}^2\) with given supports of polynomials, and degree deg \(f := \max(\operatorname{deg} f_1, \operatorname{deg} f_2)\). We show that the (non-) properness of maps \(f\) in this family depends uniquely on the pair of supports, and that the set of isolated points in \(\mathbb{C}^2 \setminus f (\mathbb{C}^2)\) has a size of at most 6 deg \(f\). This improves an existing upper bound \((\operatorname{deg} f - 1)^2\) proven by Jelonek. Moreover, for each \(n \in \mathbb{N} \), we construct a dominant map \(f\) as above, with \(\operatorname{deg} f = 2 n + 2\), and having \(2n\) isolated points in \(\mathbb{C}^2 \setminus f (\mathbb{C}^2)\). Our proofs are constructive and can be adapted to a method for computing isolated missing points of \(f\). As a byproduct, we describe those points in terms of singularities of the bifurcation set of \(f\).