  We introduce the imaginary projection of a multivariate polynomial $f \in
\mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its
imaginary part, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \,
\mathbf{z} \in \mathcal{V}(f) \}$. Since a polynomial $f$ is stable if and only
if $\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$, the notion offers a
novel geometric view underlying stability questions of polynomials.
  We show that the connected components of the complement of the closure of the
imaginary projections are convex, thus opening a central connection to the
theory of amoebas and coamoebas. Building upon this, the paper establishes
structural properties of the components of the complement, such as lower bounds
on their maximal number, proves a complete classification of the imaginary
projections of quadratic polynomials and characterizes the limit directions for
polynomials of arbitrary degree.
