I want to know if I study mechanical engineering could I end up as a product Designer or is there such a field related to mechanical engineering like a branch off of it?
What does being a “Designer” mean to you?
Yes, you can. In the ads, Mech Eng Design often means HVAC design which is not product design so it is harder to find jobs where the ME is involved in the creative side of product development. There are very few Masters programs in product design programs geared for ME’s. Stanford has one. Carnegie Mellon has one. Other schools typically think that an ME who is interested in Product Design mean you want to focus on the manufacturing of the product, which is important but that may not be what you are interested in.
If it is your idea to be involved in the actual design of new products, then there is a certain amount of industrial design you want incorporated in your education either by taking classes directly or getting the experience through other means. So what is it your envision yourself doing?
MEs design in all sorts of realms, not just HVAC. Many who ask that question really don’t have an idea of what they are asking though, hence why I asked the question. I think they don’t realize the complexity of the things we take for granted today. For example, “I want to design cool stuff like the iPhone” is typically asked by someone who doesn’t realize there are MANY designers, not just one. The camera alone has more than 400 engineers, many of whom are working on design of components. Alas, no reply from the OP. By design are we talking the look and feel of something, or the design of the integral parts of things?
@eyemgh I was only talking about when you look in the ad for jobs as “Mechanical Engineer Designers”, most of the hits are for HVAC design having spent a few months helping my son look 
There are other design jobs for MEs but they require hunting. Searching Product Designers gets you a lot of hits for industrial designers so again, you have to hunt.
I’m not sure where you are searching but it’s apparently the wrong place or the wrong search term. Really, the majority of ME no se Are involved in some part of the design process. It’s just a matter of which part. They won’t necessarily say “design” in the ad, though.
ME can do design also for cars (automotive). Here is a link if OP is interested…https://en.wikipedia.org/wiki/Automotive_design
This is what I mean: https://johnsonmirmiranthompson-openhire.silkroad.com/epostings/index.cfm?fuseaction=app.dspjob&jobid=1471&company_id=16357&version=1&jobBoardId=1112
ME also can do design for planes. But, the jobs will list as aerospace engineer not aerospace designer. This is one example as follows: Position requires aerodynamic design and analysis of missiles and fixed-wing aircraft using theoretical, semi-empirical, and CFD methods.
The job is listed as : Aerospace Engineer (by Dynatics-Huntsville, AL)
Look into the field of Industrial Design. A lot of IDs I know are also MEs (and visa versa). Schools like Carnegie Mellon (my alma mater) have strong programs in it. http://design.cmu.edu/content/products
odannyboySF: that design program from Carnegie Mellon. is it under Engineering Dept (school of engineering)?