P, NP and specialised Turing Machines
Algorithms and machines are not defined in your question and I don't think they are needed to ask what you want to ask.

Complexity classes are defined using Turing machines. That is their definition. If you want to prove anything you have to use these definitions. Anything about any other model is unrelated unless you prove some correspondence between that model and Turing machines.

Let me add that there is a hypothesis that says that "efficient" "computation" in any "reasonable" "machine" will capture the same number theoretic functions. However it is not a provable statement unless one defines the quoted terms. We can prove it for many machines but not all machines. Being Turing machine equivalent is not sufficient, we want them to be Strongly so, i.e. we should be able to simulate those machines with Turing machines and vice versa efficiently. The nice thing about $\mathsf{P}$ and $\mathsf{NP}$ is that they are very robust classes, i.e. small and big differences in machine models does not change the class. However it is true all the time. For example, I can define a simple new model of computation where I have an basic operation that in constant time solves a problem that is unsolvable in $\mathsf{P}$. Then it is obvious that this model will not be strongly equivalent to Turing machines. 

The example above was an artificial one. However right now we have a model of computation that seems closer to efficient computation in practice than polynomial-time Turing machines: bounded-error probabilistic/randomized Turing machines, efficient algorithms in that model are referred to as $\mathsf{BPP}$.

Many experts believe that $\mathsf{BPP}=\mathsf{P}$ but we are far from proving it. 

We have another model of computation which some experts expect to become practicable in future: quantum Turing machines, efficient algorithms in that model are referred to  as $\mathsf{BQP}$. We don't know if it is different from $\mathsf{P}$ but some experts conjecture that it is different and more powerful than $\mathsf{P}$.



In short, if you want to prove a complexity theory result about $\mathsf{P}$ and $\mathsf{NP}$, you have to use either the original definitions using Turing machines, or first prove a correspondence between them and the classes you are using based on some other model of computation. As I mentioned above it is quite easy to create machine models where polynomial "time" in them can solve an $\mathsf{NP}$ (or even arbitrarily more difficult) problem: just add a basic operation to your machine model that solves that problem and let the "time" that operation takes to be one unit of time.