5. Given that $A B C D-A_{1} B_{1} C_{1} D_{1}$ is a cube with edge length 1, $P$ is a moving point on the line segment $A B_{1}$, and $Q$ is a moving point on the base $A B C D$. Then the minimum value of $P C_{1}+P Q$ is ( ).
(A) $1+\frac{\sqrt{2}}{2}$
(B) $\sqrt{3}$
(C) 2
(D) $\frac{1}{2}+\frac{\sqrt{5}}{2}$
5. (A).

As shown in Figure 2, $P$ is any point on $A B_{1}$, and $Q$ is a moving point on the base $A B C D$. It is easy to see that when point $Q$ is the projection of $P$ on $A B$, $P Q$ is minimized (the leg of a right triangle is less than the hypotenuse). Let
$$
A Q=x(0<P C_{1}>B_{1} C_{1}=1).
$$
Therefore, $y \geqslant 1+\frac{\sqrt{2}}{2}$.
When the equality holds, we find that $x=1-\frac{\sqrt{2}}{2}$ satisfies $0<x<1$. Thus, $y_{\text {min }}=1+\frac{\sqrt{2}}{2}$.