While I was on base leg the other aircraft announced that he was also on base leg from a "straight in join on base", and as he was lower than me
at that point in the circuit you should both have been at the same height; either you were too high or he was too low!
I made a decision to go around and fly another circuit.
Where did you initiate the go around, base leg, downwind or on final? Many pilots don't seem to realise you can go around from anywhere in the circuit, which is usually more efficient than extending downwind in an attempt to fit in.