Time complexity of function vs return value
Here is a correct version of your original problem. 


  Find the time complexity of an algorithm which takes $T(n)$ time/operations to finish when the input size is $n$, where $T(n)$ satisfies the below initial condition,
  
  $$T(1) = C$$
  and the below recurrence relation,
  $$T(n) = 2T(n/2)+C\ \text{ if } n&gt;1$$
  for some constant $C&gt;0$.  


Another correct version would be "Find the asymptotics in term of big $O$ or $\Omega$ notation for the function $T(n)$ that ...".

My answer is based on the above editions of the your original problem.

Your question and much more beyond have been answered in more generality in this reference question and answer and in particular, by the first case of the master theorem.

Assume throughout this answer and the question that $n$ is a positive integer and "/" is the integer division when it is applied to two positive integers as in Python/Java/C/C++/C#. Without these assumptions, $T(1.5)$ and $T(3)$ may not be defined.

Here is a correct computation.
Suppose $2^h\le n\lt 2^{h+1}$ for some nonnegative integer $h$.
\begin{align}
T(n) &amp; = 2T(n/2) + (2^1-1)C\\
     &amp; = 2(2T(n/4) +C)+ (2^1-1)C\\ 
     &amp; = 2^2T(n/2^2) + (2^2-1)C\\ 
     &amp; \cdots\\ 
     &amp; = 2^hT(1) + (2^h-1)C\\  
     &amp; = (2^{h+1}-1)C\\ 
     &amp; = \Omega(n) 
\end{align}
The last equality stands from the fact that $Cn\le (2^{h+1}-1)C\lt 2Cn$.