I am trying to visualize the $SU(3)$ group used in quantum field theory. I have a (reasonably) good understanding of $SU(2)$ as the double cover of $SO(3)$ and also that this is homeomorphic to $S^3$. I have also read from other questions here that $SU(3)$ is "something like" $S^5\times SU(2)$ (would that make it "something like" $S^5\times S^3$?) but I am not sure if the "something like" entails homeomorphism or if it is some more complicated bundle.
Is there a way to visualize $SU(3)$ as a real manifold (ignoring its group structure if necessary to focus on its properties as a manifold)?
The special unitary groups fit into a sequence of fiber bundles
$$SU(n-1) \to SU(n) \to S^{2n-1}$$
coming from their actions on the unit spheres of $\mathbb{C}^n$ equipped with the standard inner product. For $n = 2$ we get a fiber bundle
$$SU(2) \to SU(3) \to S^5$$
exhibiting $SU(3)$ as a (nontrivial) $S^3$ bundle over $S^5$. This bundle splits rationally in the sense that $SU(3)$ is rationally homotopy equivalent to $S^3 \times S^5$; in particular it has the same rational homology, cohomology, and rational homotopy groups as $S^3 \times S^5$, and so in particular its Betti polynomial is
$$\sum b_i(SU(3)) t^i = (1 + t^3)(1 + t^5).$$
See also this MO question.