how2girl is right, ducted is better. Ducted extracts better and with less maintenance. But it might be difficult or impossible to do, for example in a two storey house with the kitchen in the middle so that there is nowhere for the ducting to go. suzineedsahouse is right, different people have different requirements and what is important for one person might not be important for another person. I have seen a house where a window beside the cooktop was enough for the owner (I did wonder how that works on windy or rainy days). Some people are happy enough with a non-ducted rangehood. Some people insist on a ducted rangehood and are willing to rearrange their houseplan if necessary to ensure they have space to install one. I have heard of people who have chosen commercial rangehoods for their home. I know of people who have a second kitchen outside in their alfresco area which they prefer to use for some of their cooking and sometimes even those have a ducted rangehood. (Btw as far as I know a gas cooktop only affects your choice in regards to how high above the cooktop the rangehood must be.)

looks like a hard one to solve.. I'd build a proper enclosed room by extending the roof out (new roof lower than big roof.. make sure you've got proper drainage down pipe etc so you're not getting run offs from the big roof you can block that afternoon sun by building up that half wall to where that pot is? then use shade sails to block the sun from above .. we have same problem as you in the end got louvres in then going to install privacy screens to block the sun further

You can get slim range hoods that generally fit into the wall cabinets ours were approx 40cm deep which housed the complete operation, our plumber ran a 6 or so inch flexable ducting into the roof cavity and out under the eaves. Most brands make these. A photo would be helpful. https://www.thegoodguys.com.au/omega-52cm-undermount-rangehood-oru50x?CAWELAID=620013790000422290&CAGPSPN=pla&CAAGID=21711678566&CATCI=pla-376401441757&gclid=CjwKCAjw0qLOBRBUEiwAMG5xMNN63iJr0n0Bnxf31-Gt0t5s0vERt_lOwOYkC6uzovf9TDWwawry9RoCgxQQAvD_BwE&gclsrc=aw.ds&dclid=CLOv9erawdYCFc0slgodE5oLHg Where we didn't have wall cabinets we had tall boxes of stainless look laminate built up to the ceiling which contained the extraction unit, which was approx 30-40cm's and extraction ducting cheers

I was looking at today's local real estate listings and coincidentally one shows a kitchen with something on the wall above the cooktop and below the window. I've never seen something like that before. Is there such a thing as a through the wall kitchen exhaust fan. Btw from what I've heard other exhaust fans aren't really supposed to be used in kitchens because they are designed only for steam and there's no way to clean any greasy build-up.