Here one might wonder if pure quantum states could be mixed not with mere probabilities, but with something more general like amplitudes?
The superposition a|0> + b|1> describes a linear combination of the two solutions |0> and |1> to some differential equation, but there is no notion of "mixture". A good analogy to understand superpositions is the solutions to simple harmonic motion (like mass attached to a spring). If you start the system from stretched state and zero velocity, it will oscillate like cos(ωt). If instead you give it a "kick" at zero displacement, it will oscillate like sin(ωt). Since any other combination is possible (e.g. kick while stretching), the most general solution to the equation of motion of the mass spring system is acos(ωt) + bsin(ωt), where a and b are the amplitudes. In practice we usually write acos(ωt) + bsin(ωt), so there is nothing "fancy" going on for superpositions: they are just linear combinations of the possible states. (You don't hear anyone talking about a mass-spring system oscillating like cos and like sin at the same time, do you?)
Off topic note for completeness: in physics class we rewrite acos(ωt) + bsin(ωt) as A*cos(ωt-φ), but it's the so you don't see the cos and sin separately, but they are there.
As for "mixtures" those represent our state of knowledge, or rather ignorance. The mixture of 50% |0> and 50% |1> is represented as a density matrix, which has the same "statistics" under measurement. It's hard to do matrices in plain text so I'll cut if off here... but you can look it up.
EDIT: As another comment pointed out I was wrong about this, please ignore.
It seems true at least if we accept that a pure state is a mixture of itself with nothing else.
Any quantum state CAN be represented by a density matrix. That's what density matrices are for. That quantum state may be pure or may be a mixture. (A mixture density matrix could also be a partial trace representing a subsystem of a larger system instead of a true mixture of pure states.)