Why do we use 2 transistors for each path of a MUX in CMOS?
Doesn't the circuit work without them as well?


No it does not, the PMOS + NMOS circuit is called a passgate or transmission gate. So this circuit has two passgates, C0 and C1.

In order to be able to conduct a high signal, the PMOS is needed (then the NMOS does nothing).

In order to conduct a low signal, the NMOS is needed (and the PMOS does nothing).

Only when a signal is "in the middle" both NMOS and PMOS conduct the signal. Note that in digital (logic) circuits the signal should never be "in the middle".

As we do not know if the input signal is either low or high, we have to support both low and high so an NMOS and a PMOS are needed.

For a more detailed explanation of the pass gate, look here.