When defining the tangent bundle of a $n$-dimensional manifold $M$ whose coordinate-change mappings are holomorphic, do I need to specify where M is immersed? That is, I must assume that $M\subset\mathbb{C}^k$, $k>n$, so that the tangent bundle $$ TM:=\left\{ (p,v)\in M\times\mathbb{C}^k | p\in M, v\in T_pM\right\} $$
I know that this is so in the real case (that is, when $M$ is a $n$-dimensional real manifold); is there any change in the complex case?
No, not in the complex case and not in the real case, either. A manifold's tangent bundle is a well-defined object independent of any possible embedding of the manifold into another space (though an embedding into a vector space does induce a tangent bundle).
The definition of the tangent bundle you quoted also doesn't involve any embedding, in fact it could read $(p,v) \in M \times \mathbb{C}^n$ there: the tangent space has the dimension of the manifold, not a bigger one like you might need for embedding the manifold.