In the scenario you described, where Alice is moving past Bob, both Alice and Bob will observe the other person's clock running slower due to the phenomenon of time dilation in special relativity. This situation is known as the "twin paradox."
The key to understanding the twin paradox is recognizing that it involves a difference in reference frames. Each observer has their own frame of reference from which they make their measurements and observations. From Alice's perspective, she is at rest in her own reference frame, and she sees Bob moving relative to her. Likewise, from Bob's perspective, he is at rest in his own reference frame, and he sees Alice moving relative to him.
When they compare their clocks after a period of relative motion, they will both observe the other person's clock to have ticked more slowly. This can be explained by the fact that each observer's reference frame is valid, and time dilation occurs due to the relative velocity between the two frames. Both Alice and Bob can claim to be at rest in their own frames and observe the other person's clock running slower.
The apparent paradox is resolved when they reunite and compare their clocks again. At this point, it is necessary to consider the acceleration and deceleration involved in the process. The twin who experienced acceleration (in this case, Alice) will have experienced a change in their inertial frame, which breaks the symmetry. When they compare their clocks, the twin who underwent acceleration will have experienced less time passing compared to the twin who remained in a constant inertial frame.
In summary, both Alice and Bob can observe the other person's clock running slower due to time dilation, but the resolution of the paradox comes when the effects of acceleration are taken into account.