  We investigate the quantum twistor bundle constructed as a $U(1)$-quotient of
the quantum instanton bundle of Bonechi, Ciccoli and Tarlini. It is an example
of a locally trivial noncommutative bundle fulfilling conditions of the
framework recently proposed by Brzezi\'nski and Szyma\'nski. In particular, we
give a detailed description of the corresponding $C^*$-algebra of 'continuous
functions' on its noncommutative total space. Furthermore, we analyse a
different construction of a quantum instanton bundle due to Landi, Pagani and
Reina, find a basis of its polynomial algebra and discover an intriguing and
unexpected feature of its enveloping $C^*$-algebra.
