Pulse shaping and Baseband filtering
Your complex baseband signal before modulation is given by

$$s(t)=\sum_{m=-\infty}^{\infty}a_mh(t-mT)\tag{1}$$

where $a_m$ are the complex symbols, $h(t)$ is the impulse response of the transmit filter, and $T$ is the symbol period. As you correctly assumed, $h(t)$ is usually real-valued, so you need to filter the real and imaginary parts of your symbols with the same filter $h(t)$ to get the complex baseband signal $(1)$:

$$s(t)=\sum_{m=-\infty}^{\infty}\text{Re}\{a_m\}h(t-mT)+j\sum_{m=-\infty}^{\infty}\text{Im}\{a_m\}h(t-mT)\tag{2}$$

There is no need to modify the real-valued impulse response $h(t)$.