The equation $\dot{z} = e^{i \alpha} z + e^{i \varphi} z |z|^2 + b \bar{z}^3$ models a map near a Hopf bifurcation with 1:4 resonance. It is a conjecture by V. I. Arnol'd that this equation contains all versal unfoldings of ${\Bbb Z}_4$-equivariant planar vector fields. We study its bifurcations at $\infty$ and show that the singularities of codimension two unfold versally in a neighborhood. We give an unfolding of the codimension-three singularity for $b=1$, $\varphi=3\pi/2$ and $\alpha=0$ in the system parameters and use numerical methods to study global phenomena to complete the description of the behavior near $\infty$. Our results are evidence in support of the conjecture.