Krull dimension of $\mathbb{C}[x,y,z,w]/(xw-yz)$

There's a theorem to the effect that if $f(X_1,\ldots,X_n)$ is a non-zero polynomial over a field $K$ then $K[X_1,\ldots,X_n]/f(X_1,\ldots,X_n)$ has Krull dimension $n-1$. It's a special case of the theorem that if $R=K[X_1,\ldots,X_n]/I$ for some ideal $I$, then the Krull dimension of $R$ is the transcendence degree of the field of fractions of $R$ over $K$.

To find a chain of prime ideals in $R=K[X_1,\ldots,X_n]/f(X_1,\ldots,X_n)$ think geometrically and find a solution $(a_1,\ldots,a_n)$ of $f(X_1,\ldots,X_n)=0$ and consider the chain $(0)\subseteq(X_1-a_1)\subseteq(X_1-a_1,X_1-a_2)\subseteq\cdots \subseteq(X_1-a_1,X_1-a_2,\ldots,X_{n-1}-a_{n-1})$. If you choose $(a_1,\ldots,a_n)$ carefully this will be a suitable chain of prime ideals.

Note that $$B\cong\frac{\Bbb C[x,y,w,z]}{(x,y,xw-yz)}.$$ There is a convenient alternative way to write the ideal $(x,y,xw-yz)$ which makes the structure of $B$ more evident.