Let $U\subset\mathbb{C}$ and let $\varphi:U\to\mathbb{C}^n$ be a holomorphic injection. Is it true that there is a discrete (or better yet finite) set $Z\subset U$ such that $\varphi|_{U\backslash Z}$ is a homeomorphism onto its image?
If yes, is there a similar result for $U\subset \mathbb{C}^m$, $m\le n$ (with $Z$ analytic of dimension smaller than $m$)?
First, you could have the same issue as with immersed manifolds, say the image of a disc might "touch" itself in the limit (it loops around and touches itself). Then the subset topology of the image is not the same as the topology induced by the mapping.
But otherwise it is true that locally near every point the mapping is immersion outside of isolated points, so the image is a complex manifold there. Furthremore, locally the thing is actually a homeomorphism of a neighborhood onto a neighborhood near every point, including points where the derivative vanishes. That is, the image is a topological manifold at every point in the sense that, given any point $z \in U$ then there is a neighborhood $V$ of $z$ in $U$, such that $\varphi(V)$ is an embedded topological manifold, and $\varphi|_V$ is a homeomorphism.
How to see this? Well, locally any holomorphic map from one dimension to $n$ dimension is finite, and hence proper locally (interpreted similarly as above). So the image is a local complex analytic subvariety. Complex analytic subvarieties of dimension 1 are topological submanifolds, which can be seen using perhaps the Puiseux theorem (usually stated for two dimensions, but works for a complex one dimensional curve in any number of dimensions if you state it correctly).
Your map $\varphi$ is actually a resolution of singularities (in the sense of Hironaka, though the proof is much simpler) or normalization of the variety that is the image, usually you consider such normalizations $\varphi$ that are "mostly 1-1".
A good book to read on this is Hassler Whitney's Complex Analytic Varieties. Or perhaps Chirka's Complex Analytic Sets. I find Whitney's approach very easy to follow.