Wind noise on recorded video
Id recommend going with a Rode Mic. 

The Rode Shoe Mount Video Mic (around $250-$300) is more compact than even a short short shotgun. Secondly, there is a selector switch on the back that allows the user to select various pickup patterns, from cardiod, to stereo, directional, etc. 

Theres a few models, but check BH and Amazon. 

An entry level Shotgun would be a Sennheiser, but 1.) Its far less compact, and 2.) Price point for the capsule and power supply will run $500 and up. Lastly, youre going to be limited to XLR out without an adapter. 

The Rode Mics to be honest, sound fantastic for their price point. 

You can add an accessory Rycote Furry Windjammer to cut down the wind even further beyond the included Foam Windscreen. A Furry Windjammer will give you the best results (pricey but worth it), and last, you can add a lo-cut filter either on the Rode or in post to further reduce wind rumble.