Let $I$ be a prime ideal in $\mathbb{C}\{x_1, \ldots, x_n\}_0$ (the localization at the maximal ideal that defines $0$) and suppose that the height of $I$ is $h$. Then, there is a standard *trick* to extract a regular sequence of length $h$ from $I$. And so one can always see $V:=V(I)$ (which has codimension $h$) as an irreducible component of a complete intersection of codimension $h$.

Can you choose the regular sequence $g_1, \ldots, g_h$ so that the intersection of $V$ with each of the other irreducible components of $V(g_1,\ldots,g_h)$ is just $\{0\}$? What if we assume that the local ring of $(V(I),0)$ is Cohen-Macaulay?

It seems to me that this is a strong condition on the ideal $g:=(g_1, \ldots, g_h)$, however, you have some freedom to choose the elements within $I$. Observe, that this would imply that there exists a primary decomposition of $g$ such that the sum of the primary ideal corresponding to $V$ and any other primary ideal contains (a power of) the maximal ideal of $\mathbb{C}\{x_1, \ldots, x_n\}_0$.