By "algebraic curve" here (both in $\mathbb{R}^3$ and in $\mathbb{C}^3$), I do not only mean that the dimension of the set as an algebraic variety is 1, but also that its dimension is 1 in a topological sense (i.e., that, roughly, it is locally a continuous parametric curve).

In this context, for example, the zero set of the polynomial $x^2 +y^2=0$ in variables $x,y,z$ is a line in $\mathbb{R}^3$ if the polynomial is seen in $\mathbb{R}[x,y,z]$, but a complex variety of dimension 2 in $\mathbb{C}^3$ if the polynomial is seen in $\mathbb{C}[x,y,z]$. However, if we consider the initial line in $\mathbb{R}^3$ as the zero set of the polynomials $x=0$ and $y=0$ (which is a more natural choice of polynomials), and we see these polynomials as elements of $\mathbb{C}[x,y,z]$, then their zero set in $\mathbb{C}^3$ is a line, i.e. a complex algebraic curve. Can we apply such a procedure for any real algebraic curve?

I would also like to ask something else: Is the projection of a real algebraic curve from $\mathbb{R}^3$ to a generic plane again a real algebraic curve? In other words, is the projection the zero set of one polynomial in $\mathbb{R}[x,y]$?

I do not have an algebraic geometry background, so I would really appreciate it if you could also tell me in which books I could search to understand these things better. Most of the literature deals with complex and not real varieties.

Thank you very much!