Suppose the moon rotated on its axis just as quickly as earth. would you still always see the same side of the moon from earth?

Currently it takes about 27.3 days for the Mon to rotate full circle around it's axis. It takes about 27 days for the Moon to make full circle around the Earth. This is why we always see the same side of Moon.

If the Moon was rotating around it's axis as quickly as Earth it would mean that it takes about 24h to rotate. Currently, no matter where we stand on the Earth we can see the same side of Moon. If the Moon had 24h rotation period we would see different parts of Moon from different parts of Earth. We would see the entire surface of Moon. But, as the Moon and the Earth would have same rotation period from some point on Earth we would constantly see the same side of Moon.

Example: From Europe we would see one side, from North America other side and from Asia yet another side. But from Europe we would constantly see the same side of Moon.