I know how to check lineraly dependent and lineraly independent a set of vectors in R . I make a matrix and check its rank as it is my favourite method and I find it easy.
Now the problem is I can check these vectors over $\Bbb{C}$ by checking linear combination of these two vectors and inspecting if the scalars that I used are all zero or not
But I want to know is there a way to check that these vectors are lineraly dependent or independent by matrix method over $\Bbb{C}$ as well?
Yes, you can use the method. Consider the matrix $\begin{bmatrix}1+i & 1\\ 2i & 1+i\end{bmatrix}$, by writing the given vectors as columns of the matrix. Now, in this case, for finding the rank of the matrix, I can multiply by scalars from $\mathbb{C}$, since we need linear (in)dependence over $\mathbb{C}$.
So, multiplying the first row by $1+i$ and substracting from the second row gives the matrix $\begin{bmatrix}1+i & 1\\ 0 & 0\end{bmatrix}$ and hence the rank is $1$.
Thus the given vectors are linearly dependent.