 Solution

The domains are not conformally equivalent. If $F:\mathbb{C}\to \mathbb{H}$ were a conformal equivalence, composing it with a Cayley transform from the upper half-plane $\mathbb{H}$ to the unit disc would produce a bounded nonconstant entire function. This contradicts the Liouville theorem, proving that the complex plane is not conformally equivalent to the upper half-plane.