Though this has already been answered, I wanted to give the "geometry" of what is going on.
If we treat multiplication by $a+bi$ as a linear transformation on the space $x+yi$, then, under the usual basis of $1,i$, multiplication by $a+bi$ is written as the matrix multiplication:
In number theory, the "norm" is the determinant of this matrix. In that sense, unlike in analysis, the norm can be thought of as an area rather than a length, because the determinant can be interpreted as an area (or volume in higher dimensions.) However, the area/volume interpretation only gets you so far. The reality is that the determinant of a matrix is an "algebraic" quantity which has the nice property that it is independent of the basis chosen, so it is well defined without picking a basis.
In particular, if $A,B$ are two $n\times n$ matrices, then $\det AB = \det A \det B$, which means that the norm defined this way has the same property: $N(z)N(w)=N(zw)$. That nice property follows through to other cases of "number fields" and their rings of integers where the "area" interpretation is less clear-cut.
In particular, this "algebraic norm" is not measuring distance, but rather measuring something about the multiplicative behavior of $a+bi$. That it turns out to be the square of the geometric norm in this case is a deep geometric fact about the geometry of complex numbers.