The Soviets had some pure rocket-propelled interceptor projects...basically manned, gun-armed SAMs. There was something called the BI-1, after the two designers, Bereznyak and Isaev. (Those were their surnames, of course.)
Not very practical, however.
What could you do with rocket boosters, anyway? Solid ones might be good for take-off, but that's often not necessary for fighters. It wouldn't have been in WWII; they generally weren't too heavily laden with ordnance and they had okay takeoff rolls.
What else? Improve performance in the air? Well...mixed piston and liquid-fuelled rocket propulsion's pretty tricky, too. What are you going to do...try and make cheap Me-262 substitutes using mixed propulsion? Use the liquid-fuelled motors to climb to altitude in order to attack bomber streams, say...maybe use the motors to make high-speed attacks? The problem is that they couldn't really properly exploit the speed advantages of the Me-262 for the most part, and Germany was suffering from all kinds of materials shortages, and their metallurgy left a little to be desired in some cases...so, melty rocket boosters. Hooray. Maybe they want a really cheap high-performance fighter and build motorjet fighters, and give them mixed propulsion. That's kind of cool, at least.
(A motorjet, you ask? Well...you might know it as a "thermojet," but that's not the preferred terminology these days. There's a piston engine, but instead of spinning a propeller it drives a compressor; there's an air intake, the compressor, a combustion chamber and some other fans, so instead of a propeller you get jet exhaust. It was obsolete very quickly, but maybe it could be done on the cheap. You don't even need a propeller, really. Or maybe a pulsejet-powered fighter, even.)
Or maybe a mixed-propulsion Me-262. And there was the Me-163 and some rocket-fighter proposals and prototypes which were never operational or produced in large quantities.