It's entirely possible your child is ambidextrous, but I think the other answerers are jumping the gun a bit.
First off, 'handedness' is not a binary (or trinary) thing: it's a continuum. Some people are essentially 100% right handed, some 100% left, and some are ... mixed. I'm in that range. I'm mostly right handed - I do everything right handed that you normally would see someone do, writing, eating, bowling - but I'm capable of doing it reasonably well left-hadned also. However, I'm certainly not ambidextrous, nor am I a lefty who was trained to be a righty: I'm a righty, who happens to be more like 70% right handed.
In fact, there are really four major categories: Right, Left, Mixed, and Ambidextrous. Mixed means you have a dominant hand for activities - but the hand isn't always the same hand for different ones (my grandfather was a lefty bowler and eater, and a righty writer and drawer, for example).
On top of that, while many children will show by 3, some don't show until 5 or 6 which hand they prefer. I haven't found any evidence that we know why that is; it may be 'less dominant' handedness (where it only shows once children are taught to write using one hand) or it may be due to differences in brain development.
In terms of whether you should be worried: On balance, no, you shouldn't be worried. One to two percent of people are ambidextrous or truly mixed-handed (ie, a roughly even number of tasks done with each hand).
However, there are some studies showing that being ambidextrous makes you more likely to be at risk for some mental health disorders and other differing brain functionality, such as ADHD. This article in Live Science covers one of the studies1. It used a sample of children in Northern Finland, and found:
By age 15 or 16, mixed-handed adolescents were also at twice the risk of having symptoms of attention deficit/hyperactivity disorder (ADHD). And those ambidextrous teens with ADHD had more severe symptoms than their right-handed counterparts.
That study is somewhat controversial, as some have questioned whether the effect still exists when properly balancing for demographic factors (gender, birth weight, gestational age at birth). However, it's worth keeping in mind, as paying attention to these issues can help lead to early diagnosis and treatment. It's important to note that the issues found are relatively minor, and are not in anything close to 100% of ambidextrous children; more than likely if your child does turn out to be mixed-handed or ambidextrous, it's most significant influence will be as a party trick or a feeling of kinship with Inigo Montoya.
1Rodriguez A, et al. (2010) "Mixed-handedness is linked to mental health problems in children and adolescents". Pediatrics (doi:10.1542/peds.2009-1165).