I am currently working through Axler's Linear Algebra Done Right. From what I understand, given an infinite dimensional vector space $V$ and subspaces $U$, $W_1$, and $W_2$ of $V$ such that $V=U\oplus W_1=U\oplus W_2$, it is not necessarily the case that $W_1$ and $W_2$ are isomorphic.
Intuitively, I would think that given a basis of $U$, one could extend this to a basis of $V$, and that "basis extension" would uniquely determine the "direct-sum complement" of $U$ (up to isomorphism). Thus, I am wondering if someone could provide an example of a vector space $V$ and a subspace $U$ of $V$ such that the "direct-sum complement" of $U$ is not unique, and provide some insight into why my line of reasoning fails?
Your understanding is incorrect. If $V=U\oplus W_1=U\oplus W_2$, then $W_1$ and $W_2$ are always isomorphic. Indeed, given that $V=U\oplus W_1$, if you take the projection map $V\to V/U$ and restrict it to $W_1$, you get an isomorphism $W_1\to V/U$. So $W_1\cong V/U$, and similarly $W_2\cong V/U$, and so $W_1\cong W_2$.
(For an explicit direct isomorphism between $W_1$ and $W_2$, note that any element $x\in W_1$ can be written uniquely as $u+w$ where $u\in U$ and $w\in W_2$. The map taking $x$ to $w$ is then an isomorphism $W_1\to W_2$.)