Can dimension analysis be used in developing more advanced physics equations? It is obvious that dimensional analysis can be used to derive many classical mechanics equations (excluding constants). As long as all the dependent quantities are known. 
My question is whether this is an accepted method to derive formulas or is calculus is a must? If so can it be used in advanced areas such as Quantum mechanics and relativistic physics?
 A: Yes it can!
However, the term dimension analysis needs to be seen in its/a context.
Buckinghams Pi-Theorem did not just emerge out of the blue. And, the reason it works is not pure luck. There is a well grounded physical/theoretical basis for it. And it is the basis which allowed for dimesion analysis. Three scientists come to my mind.
Sophus Lie wrote 1888:

Each one-parameter symmetry group allows the reduction of order of the differential.

Buckingham developed his Pi-Theorem in 1914:

Any dimensionally correct relationship involving physical quantities can be expressed in terms of a maximal set of dimensionless combinations of the given quantities.

And finally, Emmy Noether in 1918:

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

Of course one could argue, that dimension analysis and the Noether Theorem / Lie Symmetry Groups have different levels of abstraction.
A: As an addendum to @CuriousOne answer, also in mathematical analysis the dimensional analysis (or more properly scale transformations) can be used to guess a priori estimates and useful results, that anyways has to be proven in a rigorous fashion by other means.
Two relevant examples may be the Sobolev and Strichartz estimates, whose admissible indices can be deduced by exploiting scaling invariance.
A: Dimensional analysis only gives results that are correct to first order and up to a constant. So really there is just some qualitative guessing on physical behaviour with dimensional analysis.
A: Dimensional analysis can help to "guesstimate" the form of many important results but it can, for instance, not produce general solutions to equations of motion. It's an invaluable tool to understand the structure of physical theory, including quantum mechanics and relativity, and to check results for consistency, but it can rarely replace complex calculations. It would be nice, though, if it could. 
