Time Complexity of Shifting
In the RAM machine model, operations on "machine words" cost $O(1)$, where a machine word has width $O(\log n)$ bits, $n$ being the length of the input (in bits). This reflects the instruction sets of real world CPUs, as mentioned in the other answers.

While the RAM machine model makes sense for many algorithmic tasks, it is not a good fit for large integer arithmetic, since in practice the machine word is fixed while the integer length is growing. Instead, one usually measures bit operations. Shifting an $m$-bit integer by $c$ bits takes $O(m+c)$ bit operations.

These issues are explored in a paper of Martin Fürer, the inventor of the fastest known integer multiplication algorithm (which is also the fastest known integer division algorithm).