Hi,
What do you mean by "stockbroking"? Do you mean the sale of stocks to retail clients or actually being in ECM (Equity Capital Markets) arranging IPOs and dealing with corporate finance?
For retail broking, any degree will do really. Many people graduate in different degrees, but if you have a passion for the markets, the firm and a basic knowledge of how markets work (like excess demand creates higher prices, some sectors are more volatile than others etc, but they will teach what market cap, P/E, short-selling, chart reading etc). Many graduates in the U.S. have only studied BA and got in easy to stockbroking. But you do need fluent communication skills, able to work in high pressure environments, good numerical skills.
For ECM on the other hand, you usually need a business type degree so in if you planning to work in Australia a BCom will do just fine. It's more complex as you need to deal with models, do bookbuilds (for IPOs) etc. It's more competitive to get into and does require a solid understanding in accounting, maths and indeed economics. However, if you research, network, attend and apply for leadership positions at clubs, get a mid-D (H2A) average, show how you effectively deal in difficult situations you should be good.