If you take a quaternion number lets call it Q (Q=a+bi+cj)) Q² definitely generate a pythagorean quadruple and Q=a+bi+cj+dk will definitely generate pythagorean quintuple
ex: Q=1+1i+1j+k, Q² = (-2)+2i+2j+2k, and 2²+2²+2²+2²=16 and 16 is 4² and this is true for any integers a,b,c,d
proof 1: Q = a+bi+cj+dk, Q² = (a+bi+cj+dk)(a+bi+cj+dk)= (a² - b² - c² - d²) + 2abi + 2acj + 2adk remove all the complex numbers so it'll be (a² - b² - c² - d²) + 2ab + 2ac + 2ad square every part so it'll be (a² - b² - c² - d²)² + (2ab)² + (2ac)² + (2ad)² and if you factor it it'll be (a²+b²+c²+d²) and there is it 4 squared numbers that equal another square