I have implemented a Least Squares Optimization with $ {L}_{1} $ Norm Regularization for the problem. My reference was - Least Squares Optimization with L1 Norm Regularization by Mark Schmidt.
The paper discuss the problem of solving:
$$ {\left\| A x - b \right\|}_{2}^{2} + {\left\| x \right\|}_{1} $$
The paper assume all variables are under the real domain. Namely $ A \in \mathbb{R}^{m \times n}, \; x \in \mathbb{R}^{n}, \; b \in \mathbb{R}^{m} $.
How would one solve the problem over the complex domain where $ A \in \mathbb{C}^{m \times n}, \; x \in \mathbb{C}^{n}, \; b \in \mathbb{C}^{m} $?