The angular momentum of a particle rotating about a point is given by $\vec{L} = \vec{r} \times \vec{p}$.
Imagine a particle tracing a circular path on a flat table. If I put the origin of my coordinate system at the center of the circular path such that the flat table is the $xy$-plane, the angular momentum vector points in the positive direction of the $z$-axis because $\vec{r}$ lies on the xy-plane and $\vec{p}$ is always perpendicular to $\vec{r}$. Therefore, there is no change in $\vec{L}$ throughout the motion of the particle.
Now, I only move the origin vertically toward the ground. Now $\vec{r}$ makes a certain angle $\phi$ with the $z$-axis, and therefore, $\vec{L}$ also makes the same angle $\phi$ with the $z$-axis. But, now $\vec{L}$ keeps changing its direction as the particle is moving on the circular path on the flat table!
To conclude, an angular momentum $\vec{L}$ depends so much on the choice of the origin pretty much unlike the linear momentum $\vec{p}$?