How to design a computer-based multiple-choice quiz?
What you describe is pretty much exactly what we do with final exams in our large-section math courses at my institution. For a quantitative-heavy subject like ours, the following is feasible: effectively ask everyone the same questions but with different numerical values involved. 

Our elementary algebra course uses the Maple T.A. platform, where question formats can be coded and somewhat different numerical values presented to each student. 

Our remedial arithmetic course uses the Blackboard testing system, which does not support randomizing values for fraction or radical-form answers (but could for decimal-form answers: see questions of type Calculated Formula), so the faculty have manually drafted about a half-dozen separate, parallel tests of equivalent difficulty. 

The Maple product is definitely a better system for this kind of setup, but it may cost money, whereas Blackboard is potentially already on site. In each case, the order of questions, and the order of responses, can be randomized for each student. 

The overall process works tolerably well. Downsides: To my mind, the questions are rather obviously functionally "the same" for every student -- but the students (at least at this level) don't seem to perceive them this way. Also, because the overall scope of the questions is fixed, in some sections it leads to rather degenerate teaching-to-the-test, and students in the next course in the sequence don't really have any understanding or depth with the skills in question. (For example: Some instructors only teach checking multiple-choice solutions to equations, not actually solving them, etc.) In any case, quite a bit of work is involved setting up your tests.