Uncomputably coded model of computation
Consider a standard coding of Turing machine $T_0$, $T_1$, $T_2$, ... Let $b : \mathbb{N} \to \mathbb{N}$ be a non-computable bijection. Define a new coding of Turing machines $T'_n = T_{b(n)}$. Clearly, the encoding $T'$ can do everything that Turing machines can do (whatever you do with $T$ you can do with $T'$ by composing with $b$ and its inverse), but it is not equivalent to $T$ by a computable bijection (or else $b$ would be computable).