Complex cobordism ring of the quaternionic projective plane ($M{U}^{\ast }({\mathbb{HP}}^2)\cong M{U}^{\ast }(\mathrm{pt})[q]/(q^3),|q|=4$)

For the quaternionic tautological line bundle regarded as a complex rank-two bundle, take $q=c_2^{MU}$. Its ordinary image generates degree-four integral cohomology, and its square generates degree eight. The Atiyah-Hirzebruch spectral sequence collapses because the cells have dimensions $0,4,8$ and the coefficients are even. The classes $1,q,q^2$ lift a free basis of the associated graded module, and $q^3=0$ because the theory is connective and the space has dimension eight.