I am solving the following problem: $x\in\mathbb{H}$ where $x=a+bi+cj+dk$ and $a,b,c,d\in \mathbb{R}$ is called a pure quaternion if and only if $a=0$. Show that there are infinitely many pure quaternions that satisfy $x^2+1=0$.
My work so far: Let $x=bi+cj+dk$ since if $x=0$, then $1\neq 0$ and $0$ is not a solution. $x^2=-b^2-c^2-d^2$ by computation $\implies b^2+c^2+d^2=1$ which is the equation for the unit sphere. Intuitively, I can see that there are infinitely (in fact, uncountably) many solutions, but how do I prove that is the case?