  We study the topological configurations of the two-component condensates of
bosons with the $3$D $\vec{\sigma}\cdot \vec{p}$ Weyl-type spin-orbit coupling
subject to a harmonic trapping potential. The topology of the condensate
wavefunctions manifests in the quaternionic representation. In comparison to
the $U(1)$ complex phase, the quaternionic phase manifold is $S^3$ and the spin
orientations form the $S^2$ Bloch sphere through the 1st Hopf mapping. The
spatial distributions of the quaternionic phases exhibit the 3D skyrmion
configurations, and the spin distributions possess non-trivial Hopf invariants.
Spin textures evolve from the concentric distributions at the weak spin-orbit
coupling regime to the rotation symmetry breaking patterns at the intermediate
spin-orbit coupling regime. In the strong spin-orbit coupling regime, the
single-particle spectra exhibit the Landau-level type quantization. In this
regime, the three-dimensional skyrmion lattice structures are formed when
interactions are below the energy scale of Landau level mixings. Sufficiently
strong interactions can change condensates into spin-polarized plane-wave
states, or, superpositions of two plane-waves exhibiting helical spin spirals.
