Define a category as follows: the objects are categories and the arrows are embeddings (full and faithful functors). Is this category Cauchy complete or complete?
It is easy to see that every idempotent splits in your category: if $e:C\to C$ is idempotent and fully faithful, let $B$ be the full subcategory of $C$ spanned by the objects in the image of $C$. Then the inclusion $B\to C$ is fully faithful, and $e$ factors fully faithfully through this inclusion, giving a splitting of $e$.
However, your category is not complete, because it does not have products. For instance, let $C$ be the discrete category with two objects $a$ and $b$. Suppose you have a product $P=C\times C$, with projection functors $p_0,p_1:P\to C$. There are two fully faithful functors $C\to C$, namely the identity $1$ and the functor $f$ which swaps $a$ and $b$. There is a unique fully faithful functor $g:C\to P$ such that $p_0g=p_1g=1$, and also a unique fully faithful functor $h:C\to P$ such that $p_0h=1$ and $p_1h=f$. Since the $p_i$ are fully faithful, $p_0g=p_0h=1$ implies $g(a)\cong h(a)$ and $g(b)\cong h(b)$ in $P$. But $p_1g=1$ and $p_1h=f$ implies $g(a)\cong h(b)$ and $g(b)\cong h(a)$. This implies $g(a)\cong g(b)$, contradicting that $g$ should be fully faithful.