Confusion in Second Derivative of a Vector Valued Function

Question

I have a slight confusion in Newton Raphson method applied in Higher Dimensions. I have a vector valued function $f(x)$ which maps from $\mathbb{C^{m}}\rightarrow \mathbb{C^{m}}$ whose root I have to find. I resorted to Newton-Raphson first order method to do this, namely: $${\bf{x_{n+1}}}={\bf{x_{n}}}-\frac{f(x)}{\dot{f(x)}}$$ However, I started facing problems since sometimes the linear approximation of the function $f(x)$ was quite poor and hence tended to diverge or oscillate. To fix this, I have to introduce some sort of scaling for the Newton step and HouseHolder's second order iterate seems like a suitable candidate. Its given by:$${\bf{x_{n+1}}}={\bf{x_{n}}}-\frac{f(x)}{\dot{f(x)}}\left(1+\frac{f(x)\ddot{f(x)}}{2\dot{f(x)}^{2}}\right)$$ I understand that $\dot{f(x)}$ is the ${\bf{Jacobian}}$. However, I am quite unsure of what $\ddot{f(x)}$ means for a vector valued function(it would be the ${\bf{Hessian}}$ for a multivariate function mapping to $\mathbb{R}$) and hence I am unable to compute it. Could someone please clarify?