  We study the class of quantum measurements with the property that the image
of the set of quantum states under the measurement map transforming states into
probability distributions is similar to this set and call such measurements
morphophoric. This leads to the generalisation of the notion of a qplex, where
SIC-POVMs are replaced by the elements of the much larger class of morphophoric
POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The
intrinsic geometry of a generalised qplex is the same as that of the set of
quantum states, so we explore its external geometry, investigating, inter alia,
the algebraic and geometric form of the inner (basis) and the outer (primal)
polytopes between which the generalised qplex is sandwiched. In particular, we
examine generalised qplexes generated by MUB-like 2-design POVMs utilising
their graph-theoretical properties. Moreover, we show how to extend the primal
equation of QBism designed for SIC-POVMs to the morphophoric case.
