Generalization of quantum computing ?

This is not particularly related to quantum computing itself but rather to the notion of extending the manipulating a single value to manipulating a linear combination of such values, having the combinations denote the amplitude of measurement of a single value. One extension approach would be to consider non-linear combinations, but that's not what I mean here. I am asking what if this was generalized somehow to the linear combinations of such linear combinations ? What is the possible operations we could do in a manner similar to that of quantum computing and what useful outcomes would that mean to us ? Quantum computing allowed things not thought to be possible like sub-linear search complexity (Grover's algorithm) and others like Shor's algorithm. What would further generalizations like the proposed one allow us to do that we didn't think was possible with quantum computing ? Another research direction I may want to pursue after PhD en shaa Allah =)