Consider a register machine which computes a function from $\mathbb N \rightarrow \mathbb N$. By computing a function it's meant that a register machine with alphabet $\{a\}$, starting with $x$ number of $a$ letters as its input will print $y$ number of $a$ letter's and halt. $x$ and $y$ are the input/output of the function.
Question, does extending the alphabet to for example $\{a,b\}$ increase the computational power of the register machine (i.e. is there a function that can be computed only with the extended alphabet)?