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The phase shift between pressure and wall shear stress (WSS) has been associated with cardiovascular diseases such as atherosclerosis and aneurysms. The present study aims to understand the effects of geometry and flow properties on the phase shift under the stiff wall assumption, using an immersed-boundary-lattice-Boltzmann method. For pulsatile flow in a straight pipe, the phase shift is known to increase with the Womersley number but is independent of the flow speed (or Reynolds number). For a complex geometry, such as a curved pipe, however, we find that the phase shift develops a strong dependence on geometry and Reynolds number. We observe that the phase shift at the inner bend of the curved vessel and in the aneurysm dome is larger than that in a straight pipe. Moreover, the geometry affects the connection between the phase shift and other WSS-related metrics, such as time-averaged WSS (TAWSS). For straight and curved blood vessels, the phase shift behaves qualitatively similarly to and can thus be represented by the TAWSS which is a widely used hemodynamic index. However, these observables significantly differ in other geometries, such as in aneurysms. In such cases, one needs to consider the phase shift as an independent quantity that may carry additional valuable information compared to well-established metrics.