can I build a multiplexer with just four transistors?
There are a lot of "tricks" to get a single stage non-inverting logic gate to work, but a lot of times they will suffer from some input or output characteristic that is undesirable. In this circuit's case, I can identify the following issues you may see (in no particular order, not exhaustive).


When Ih = Co = '1', the output voltage will be two Vbe drops lower than Co
When Il = '1', Co = '0', the output voltage will be one Vbe drop and Vsat lower than Il
Output voltage is not restorative, meaning you couldn't chain circuits using this technique one after the other forever
Output voltages differ for various conditions of Ou = 1
When Il = '1', Co = '0', Il will be shorted to Co through the base-emitter junction of Q2
When Ih = '0', Co = '1', Ih will be shorted to Co through the base-collector junction of Q1
Input impedance depends on states of other inputs (and changes dramatically)
Reliable operation may require pull-down resistors on the base of Q3 and Q4
Depending on voltage levels, reverse Vbe may be exceeded (typically -5V)


So it can probably be made to work, but there's a lot of issues with using it as a normal logic gate. You would likely need to test circuit behavior in every state of this gate, monitoring voltages and currents. If I grabbed a standard logic gate (TTL, CMOS, etc...), then I would generally expect none of these issues to really cause problems.