How does the computational complexity of problems depend on the model specifics of a register machine?
The basic definitions of computational complexity are normally phrased in terms of plain Turing Machines, so one has to be careful about transferring results to different models. Naturally, if you can perform look-ups faster, any program that requires that functionality will also be faster (at least for that part).

However if the other model is not to extreme, then the gap in the precise running times falls below our normal threshold of caring. That is, most models of fairly general interest are polynomially equivalent to Turing Machines, so even though a register machine is in some sense faster than a Turing Machine, it's not so much faster that it breaks the definitions - it only speeds things up by a polynomial factor (i.e. an $O(n^{a})$ algorithm will only go down to a $O(n^{b})$ algorithm with $b \leq a$).

You can of course define models that are significantly faster than a Turing Machine (for a trivial example, a Turing Machine that can solve $\mathsf{Clique}$ in a single step), at this point, you can't feasibly replace "Turing Machine" in the definition of $\mathsf{P}$ or $\mathsf{NP}$ with the new model without changing things significantly (but models like this do have other uses).