I have taken math studies an was wondering if i had to shift to maths HL if i wanted to do this major
It can be, if it is a specialized version of computer science.
However, game companies hire lots of general computer science majors, so a computer game design major may not be that advantageous there, but may be limiting for other computing jobs.
at least where my son is studying (RIT) it is in the college of Computing. Is it going to be a BS or BA? that is where it might differ.
What do you mean by game design? If you look at USC for an example, they have at least two programs that could be considered game design. One is a concentration in CS. That would be math heavy. The second is a minor in the School of Cinematic Arts. That one is not math heavy.
Using a local school as an example there are two tracks available for game design. One is a BFA with emphasis on creating visuals/backgrounds/characters. The other is a BS with a higher math requirement for doing more of the logic programming.
If i was going for a BS would i need to shift to higher level maths or not
If you are aiming for CS, choose a math course intended to prepare students for more math later. Among IB math, this means not math studies SL or math AI SL.
I work in game design, and if you are going for a game design course, I would recommend the higher-level math. There are far more jobs for software developers in games than there are for designers, and a lot of designers come from computer science backgrounds anyway.