  An algebraic framework for noncommutative bundles with (quantum) homogeneous
fibres is proposed. The framework relies on the use of principal coalgebra
extensions which play the role of principal bundles in noncommutative geometry
which might be additionally equipped with a Hopf algebra symmetry. The proposed
framework is supported by two examples of noncommutative $\mathbb{C}
P_q^1$-bundles: the quantum flag manifold viewed as a bundle with a generic
Podle\'s sphere as a fibre, and the quantum twistor bundle viewed as a bundle
over the quantum 4-sphere of Bonechi, Ciccoli and Tarlini.
