Question: Suppose I have two 5-digit numbers (A and B) and two 50-digit numbers(C and D). Do the operations **A+B** and **C+D** have equal complexity in terms of time and space? or **C+D** is more complex due to the size of digits?

Answer: 
The complexity of arithmetic operations is typically measured in *bit operations*, though not everybody agrees this is the correct measure. There is a [Wikipedia page](https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations#Arithmetic_functions) listing the complexity of the best algorithms currently known for various operations, including addition and multiplication.
It depends on the model you use. [Yuval Filmus's answer](https://cs.stackexchange.com/a/124164/83244) assumes you use a [multitape Turing machine](https://en.wikipedia.org/wiki/Multitape_Turing_machine). This is also mentioned in [the Wikipedia page](https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations): 

> 
> Here, complexity refers to the time complexity of performing
>  computations on a multitape Turing machine.
> 
> 
> 

However, for example, if you use a [random access machine](https://en.wikipedia.org/wiki/Random-access_machine), the time and space complexity of an arithmetic operation of any-digit numbers is considered to be $O(1)$.