A conical pendulum consists of a simple pendulum moving in a horizontal circle as shown in the figure. C is the pivot, O the centre of the circle in which the pendulum bob moves and $\omega$ the constant angular velocity of the bob. If $\overrightarrow{L}$ is the angular momentum about point C, then

The direction  of $\overrightarrow{L}$ is perpendicular to the line joining the bob at point C. Since this line keeps changing its orientation in space, direction of $\overrightarrow{L}$ keeps changing, however, as $\omega$ is constant, magnitude of $\overrightarrow{L}$ remains constant. 

Method-2: The torque about point is perpendicular to the angular momentum vector about point C. Hence it can only change the direction of L, and not its magnitude.