  The standard Radon transform of holomorphic functions is not always well
defined, as the integration of such functions over planes may not converge. In
this paper, we introduce new Radon-type transforms of co-(real)dimension $2$
for harmonic and holomorphic functions on the unit ball. These transforms are
abstractly defined as orthogonal projections onto spaces of complex harmonic
and holomorphic plane waves, respectively. The inversion formulas are derived
based on the dual transform, while the latter is defined as an integration on a
complex Stiefel manifold. Our transforms are extended to the Fock space and
give rise to a new transform defined on the entire $L^{2}(\mathbb{R}^{n})$
through the Segal-Bargmann transform. Furthermore, we develop these transforms
for Hermitian monogenic functions on the unit ball, thereby refining the
Szeg\"o-Radon transform for monogenic functions introduced by Colombo, Sabadini
and Sommen.
